author | haftmann |
Fri, 04 Jul 2014 20:18:47 +0200 | |
changeset 57512 | cc97b347b301 |
parent 56479 | 91958d4b30f7 |
child 58249 | 180f1b3508ed |
permissions | -rw-r--r-- |
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(* Title: HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy |
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Author: Amine Chaieb |
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*) |
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header{* A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008 *} |
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theory Parametric_Ferrante_Rackoff |
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imports |
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Reflected_Multivariate_Polynomial |
|
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Dense_Linear_Order |
|
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DP_Library |
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"~~/src/HOL/Library/Code_Target_Numeral" |
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"~~/src/HOL/Library/Old_Recdef" |
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begin |
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subsection {* Terms *} |
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|
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datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm |
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| Neg tm | Sub tm tm | CNP nat poly tm |
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|
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(* A size for poly to make inductive proofs simpler*) |
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primrec tmsize :: "tm \<Rightarrow> nat" |
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where |
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"tmsize (CP c) = polysize c" |
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| "tmsize (Bound n) = 1" |
26 |
| "tmsize (Neg a) = 1 + tmsize a" |
|
27 |
| "tmsize (Add a b) = 1 + tmsize a + tmsize b" |
|
28 |
| "tmsize (Sub a b) = 3 + tmsize a + tmsize b" |
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29 |
| "tmsize (Mul c a) = 1 + polysize c + tmsize a" |
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| "tmsize (CNP n c a) = 3 + polysize c + tmsize a " |
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(* Semantics of terms tm *) |
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primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" |
|
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where |
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"Itm vs bs (CP c) = (Ipoly vs c)" |
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| "Itm vs bs (Bound n) = bs!n" |
37 |
| "Itm vs bs (Neg a) = -(Itm vs bs a)" |
|
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| "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b" |
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| "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b" |
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| "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a" |
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| "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t" |
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|
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fun allpolys :: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool" |
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where |
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"allpolys P (CP c) = P c" |
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| "allpolys P (CNP n c p) = (P c \<and> allpolys P p)" |
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| "allpolys P (Mul c p) = (P c \<and> allpolys P p)" |
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| "allpolys P (Neg p) = allpolys P p" |
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| "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)" |
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| "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)" |
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| "allpolys P p = True" |
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|
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primrec tmboundslt :: "nat \<Rightarrow> tm \<Rightarrow> bool" |
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where |
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"tmboundslt n (CP c) = True" |
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| "tmboundslt n (Bound m) = (m < n)" |
57 |
| "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)" |
|
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| "tmboundslt n (Neg a) = tmboundslt n a" |
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| "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)" |
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| "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" |
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| "tmboundslt n (Mul i a) = tmboundslt n a" |
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||
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primrec tmbound0 :: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) |
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where |
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"tmbound0 (CP c) = True" |
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| "tmbound0 (Bound n) = (n>0)" |
67 |
| "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)" |
|
68 |
| "tmbound0 (Neg a) = tmbound0 a" |
|
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| "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)" |
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| "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" |
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| "tmbound0 (Mul i a) = tmbound0 a" |
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lemma tmbound0_I: |
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assumes nb: "tmbound0 a" |
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shows "Itm vs (b#bs) a = Itm vs (b'#bs) a" |
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using nb |
77 |
by (induct a rule: tm.induct,auto) |
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primrec tmbound :: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) |
80 |
where |
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"tmbound n (CP c) = True" |
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| "tmbound n (Bound m) = (n \<noteq> m)" |
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| "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)" |
|
84 |
| "tmbound n (Neg a) = tmbound n a" |
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| "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)" |
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| "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" |
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| "tmbound n (Mul i a) = tmbound n a" |
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|
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lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" |
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by (induct t) auto |
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|
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lemma tmbound_I: |
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assumes bnd: "tmboundslt (length bs) t" |
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and nb: "tmbound n t" |
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and le: "n \<le> length bs" |
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shows "Itm vs (bs[n:=x]) t = Itm vs bs t" |
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using nb le bnd |
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by (induct t rule: tm.induct) auto |
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fun decrtm0 :: "tm \<Rightarrow> tm" |
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where |
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"decrtm0 (Bound n) = Bound (n - 1)" |
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| "decrtm0 (Neg a) = Neg (decrtm0 a)" |
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| "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)" |
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| "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)" |
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| "decrtm0 (Mul c a) = Mul c (decrtm0 a)" |
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| "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)" |
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| "decrtm0 a = a" |
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fun incrtm0 :: "tm \<Rightarrow> tm" |
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where |
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"incrtm0 (Bound n) = Bound (n + 1)" |
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| "incrtm0 (Neg a) = Neg (incrtm0 a)" |
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| "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)" |
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| "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)" |
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| "incrtm0 (Mul c a) = Mul c (incrtm0 a)" |
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| "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)" |
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| "incrtm0 a = a" |
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lemma decrtm0: |
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assumes nb: "tmbound0 t" |
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shows "Itm vs (x # bs) t = Itm vs bs (decrtm0 t)" |
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using nb by (induct t rule: decrtm0.induct) simp_all |
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lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t" |
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by (induct t rule: decrtm0.induct) simp_all |
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primrec decrtm :: "nat \<Rightarrow> tm \<Rightarrow> tm" |
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where |
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"decrtm m (CP c) = (CP c)" |
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| "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))" |
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| "decrtm m (Neg a) = Neg (decrtm m a)" |
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| "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)" |
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| "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)" |
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| "decrtm m (Mul c a) = Mul c (decrtm m a)" |
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| "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))" |
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primrec removen :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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where |
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"removen n [] = []" |
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| "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))" |
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lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs" |
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by (induct xs arbitrary: n) auto |
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lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)" |
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by (induct xs arbitrary: n) auto |
148 |
||
149 |
lemma removen_length: |
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"length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)" |
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by (induct xs arbitrary: n, auto) |
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|
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lemma removen_nth: |
154 |
"(removen n xs)!m = |
|
155 |
(if n \<ge> length xs then xs!m |
|
156 |
else if m < n then xs!m |
|
157 |
else if m \<le> length xs then xs!(Suc m) |
|
158 |
else []!(m - (length xs - 1)))" |
|
159 |
proof (induct xs arbitrary: n m) |
|
160 |
case Nil |
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then show ?case by simp |
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next |
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case (Cons x xs n m) |
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{ |
165 |
assume nxs: "n \<ge> length (x#xs)" |
|
166 |
then have ?case using removen_same[OF nxs] by simp |
|
167 |
} |
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moreover |
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{ |
170 |
assume nxs: "\<not> (n \<ge> length (x#xs))" |
|
171 |
{ |
|
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assume mln: "m < n" |
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then have ?case using Cons by (cases m) auto |
|
174 |
} |
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moreover |
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{ |
177 |
assume mln: "\<not> (m < n)" |
|
178 |
{ |
|
179 |
assume mxs: "m \<le> length (x#xs)" |
|
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then have ?case using Cons by (cases m) auto |
|
181 |
} |
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moreover |
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{ |
184 |
assume mxs: "\<not> (m \<le> length (x#xs))" |
|
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have th: "length (removen n (x#xs)) = length xs" |
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186 |
using removen_length[where n="n" and xs="x#xs"] nxs by simp |
55768 | 187 |
with mxs have mxs':"m \<ge> length (removen n (x#xs))" |
188 |
by auto |
|
189 |
then have "(removen n (x#xs))!m = [] ! (m - length xs)" |
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using th nth_length_exceeds[OF mxs'] by auto |
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then have th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" |
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192 |
by auto |
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then have ?case |
194 |
using nxs mln mxs by auto |
|
195 |
} |
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ultimately have ?case by blast |
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} |
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ultimately have ?case by blast |
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} |
200 |
ultimately show ?case by blast |
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qed |
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|
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lemma decrtm: |
204 |
assumes bnd: "tmboundslt (length bs) t" |
|
205 |
and nb: "tmbound m t" |
|
206 |
and nle: "m \<le> length bs" |
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shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t" |
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using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth) |
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|
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primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" |
211 |
where |
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212 |
"tmsubst0 t (CP c) = CP c" |
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| "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)" |
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| "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))" |
|
215 |
| "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)" |
|
216 |
| "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)" |
|
55754 | 217 |
| "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" |
39246 | 218 |
| "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)" |
55754 | 219 |
|
220 |
lemma tmsubst0: "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a" |
|
41842 | 221 |
by (induct a rule: tm.induct) auto |
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222 |
|
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lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)" |
41842 | 224 |
by (induct a rule: tm.induct) auto |
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225 |
|
55754 | 226 |
primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" |
227 |
where |
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"tmsubst n t (CP c) = CP c" |
39246 | 229 |
| "tmsubst n t (Bound m) = (if n=m then t else Bound m)" |
55754 | 230 |
| "tmsubst n t (CNP m c a) = |
231 |
(if n = m then Add (Mul c t) (tmsubst n t a) else CNP m c (tmsubst n t a))" |
|
39246 | 232 |
| "tmsubst n t (Neg a) = Neg (tmsubst n t a)" |
233 |
| "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)" |
|
55754 | 234 |
| "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" |
39246 | 235 |
| "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)" |
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236 |
|
55754 | 237 |
lemma tmsubst: |
238 |
assumes nb: "tmboundslt (length bs) a" |
|
239 |
and nlt: "n \<le> length bs" |
|
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shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a" |
55754 | 241 |
using nb nlt |
242 |
by (induct a rule: tm.induct) auto |
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243 |
|
55754 | 244 |
lemma tmsubst_nb0: |
245 |
assumes tnb: "tmbound0 t" |
|
246 |
shows "tmbound0 (tmsubst 0 t a)" |
|
247 |
using tnb |
|
248 |
by (induct a rule: tm.induct) auto |
|
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249 |
|
55754 | 250 |
lemma tmsubst_nb: |
251 |
assumes tnb: "tmbound m t" |
|
252 |
shows "tmbound m (tmsubst m t a)" |
|
253 |
using tnb |
|
254 |
by (induct a rule: tm.induct) auto |
|
255 |
||
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lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)" |
55754 | 257 |
by (induct t) auto |
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258 |
|
55754 | 259 |
(* Simplification *) |
260 |
||
261 |
consts tmadd:: "tm \<times> tm \<Rightarrow> tm" |
|
55768 | 262 |
recdef tmadd "measure (\<lambda>(t,s). size t + size s)" |
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"tmadd (CNP n1 c1 r1,CNP n2 c2 r2) = |
55754 | 264 |
(if n1 = n2 then |
265 |
let c = c1 +\<^sub>p c2 |
|
266 |
in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1, r2)) |
|
267 |
else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) |
|
268 |
else (CNP n2 c2 (tmadd (CNP n1 c1 r1, r2))))" |
|
269 |
"tmadd (CNP n1 c1 r1, t) = CNP n1 c1 (tmadd (r1, t))" |
|
270 |
"tmadd (t, CNP n2 c2 r2) = CNP n2 c2 (tmadd (t, r2))" |
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"tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)" |
55754 | 272 |
"tmadd (a, b) = Add a b" |
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55754 | 274 |
lemma tmadd[simp]: "Itm vs bs (tmadd (t, s)) = Itm vs bs (Add t s)" |
275 |
apply (induct t s rule: tmadd.induct, simp_all add: Let_def) |
|
276 |
apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all) |
|
277 |
apply (case_tac "n1 = n2", simp_all add: field_simps) |
|
278 |
apply (simp only: distrib_left[symmetric]) |
|
279 |
apply (auto simp del: polyadd simp add: polyadd[symmetric]) |
|
280 |
done |
|
281 |
||
282 |
lemma tmadd_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 s \<Longrightarrow> tmbound0 (tmadd (t, s))" |
|
283 |
by (induct t s rule: tmadd.induct) (auto simp add: Let_def) |
|
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284 |
|
55754 | 285 |
lemma tmadd_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n s \<Longrightarrow> tmbound n (tmadd (t, s))" |
286 |
by (induct t s rule: tmadd.induct) (auto simp add: Let_def) |
|
287 |
||
288 |
lemma tmadd_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n s \<Longrightarrow> tmboundslt n (tmadd (t, s))" |
|
289 |
by (induct t s rule: tmadd.induct) (auto simp add: Let_def) |
|
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290 |
|
55754 | 291 |
lemma tmadd_allpolys_npoly[simp]: |
292 |
"allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmadd(t, s))" |
|
293 |
by (induct t s rule: tmadd.induct) (simp_all add: Let_def polyadd_norm) |
|
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294 |
|
55754 | 295 |
fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm" |
296 |
where |
|
55768 | 297 |
"tmmul (CP j) = (\<lambda>i. CP (i *\<^sub>p j))" |
298 |
| "tmmul (CNP n c a) = (\<lambda>i. CNP n (i *\<^sub>p c) (tmmul a i))" |
|
299 |
| "tmmul t = (\<lambda>i. Mul i t)" |
|
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300 |
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lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)" |
55754 | 302 |
by (induct t arbitrary: i rule: tmmul.induct) (simp_all add: field_simps) |
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303 |
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lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)" |
55754 | 305 |
by (induct t arbitrary: i rule: tmmul.induct) auto |
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306 |
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lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)" |
55754 | 308 |
by (induct t arbitrary: n rule: tmmul.induct) auto |
309 |
||
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310 |
lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)" |
55754 | 311 |
by (induct t arbitrary: i rule: tmmul.induct) (auto simp add: Let_def) |
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312 |
|
55754 | 313 |
lemma tmmul_allpolys_npoly[simp]: |
36409 | 314 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
55754 | 315 |
shows "allpolys isnpoly t \<Longrightarrow> isnpoly c \<Longrightarrow> allpolys isnpoly (tmmul t c)" |
316 |
by (induct t rule: tmmul.induct) (simp_all add: Let_def polymul_norm) |
|
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317 |
|
55754 | 318 |
definition tmneg :: "tm \<Rightarrow> tm" |
319 |
where "tmneg t \<equiv> tmmul t (C (- 1,1))" |
|
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320 |
|
55754 | 321 |
definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" |
322 |
where "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s, tmneg t))" |
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323 |
|
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lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)" |
55754 | 325 |
using tmneg_def[of t] by simp |
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326 |
|
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327 |
lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)" |
55754 | 328 |
using tmneg_def by simp |
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329 |
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330 |
lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)" |
55754 | 331 |
using tmneg_def by simp |
332 |
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333 |
lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)" |
55754 | 334 |
using tmneg_def by simp |
335 |
||
336 |
lemma [simp]: "isnpoly (C (-1, 1))" |
|
337 |
unfolding isnpoly_def by simp |
|
338 |
||
339 |
lemma tmneg_allpolys_npoly[simp]: |
|
36409 | 340 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
55754 | 341 |
shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" |
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342 |
unfolding tmneg_def by auto |
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343 |
|
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344 |
lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)" |
55754 | 345 |
using tmsub_def by simp |
346 |
||
347 |
lemma tmsub_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 s \<Longrightarrow> tmbound0 (tmsub t s)" |
|
348 |
using tmsub_def by simp |
|
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349 |
|
55754 | 350 |
lemma tmsub_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n s \<Longrightarrow> tmbound n (tmsub t s)" |
351 |
using tmsub_def by simp |
|
352 |
||
353 |
lemma tmsub_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n s \<Longrightarrow> tmboundslt n (tmsub t s)" |
|
354 |
using tmsub_def by simp |
|
355 |
||
356 |
lemma tmsub_allpolys_npoly[simp]: |
|
36409 | 357 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
55754 | 358 |
shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" |
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359 |
unfolding tmsub_def by (simp add: isnpoly_def) |
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360 |
|
55754 | 361 |
fun simptm :: "tm \<Rightarrow> tm" |
362 |
where |
|
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363 |
"simptm (CP j) = CP (polynate j)" |
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|
364 |
| "simptm (Bound n) = CNP n (1)\<^sub>p (CP 0\<^sub>p)" |
41821 | 365 |
| "simptm (Neg t) = tmneg (simptm t)" |
366 |
| "simptm (Add t s) = tmadd (simptm t,simptm s)" |
|
367 |
| "simptm (Sub t s) = tmsub (simptm t) (simptm s)" |
|
55754 | 368 |
| "simptm (Mul i t) = |
369 |
(let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')" |
|
370 |
| "simptm (CNP n c t) = |
|
371 |
(let c' = polynate c in if c' = 0\<^sub>p then simptm t else tmadd (CNP n c' (CP 0\<^sub>p ), simptm t))" |
|
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372 |
|
55754 | 373 |
lemma polynate_stupid: |
36409 | 374 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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|
375 |
shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a)" |
55754 | 376 |
apply (subst polynate[symmetric]) |
377 |
apply simp |
|
378 |
done |
|
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379 |
|
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380 |
lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t" |
55768 | 381 |
by (induct t rule: simptm.induct) (auto simp add: Let_def polynate_stupid) |
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382 |
|
55754 | 383 |
lemma simptm_tmbound0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)" |
384 |
by (induct t rule: simptm.induct) (auto simp add: Let_def) |
|
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385 |
|
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386 |
lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)" |
55754 | 387 |
by (induct t rule: simptm.induct) (auto simp add: Let_def) |
388 |
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389 |
lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)" |
55754 | 390 |
by (induct t rule: simptm.induct) (auto simp add: Let_def) |
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391 |
|
55754 | 392 |
lemma [simp]: "isnpoly 0\<^sub>p" |
393 |
and [simp]: "isnpoly (C(1,1))" |
|
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394 |
by (simp_all add: isnpoly_def) |
55754 | 395 |
|
396 |
lemma simptm_allpolys_npoly[simp]: |
|
36409 | 397 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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398 |
shows "allpolys isnpoly (simptm p)" |
55754 | 399 |
by (induct p rule: simptm.induct) (auto simp add: Let_def) |
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400 |
|
41822 | 401 |
declare let_cong[fundef_cong del] |
402 |
||
55754 | 403 |
fun split0 :: "tm \<Rightarrow> (poly \<times> tm)" |
404 |
where |
|
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405 |
"split0 (Bound 0) = ((1)\<^sub>p, CP 0\<^sub>p)" |
55754 | 406 |
| "split0 (CNP 0 c t) = (let (c', t') = split0 t in (c +\<^sub>p c', t'))" |
407 |
| "split0 (Neg t) = (let (c, t') = split0 t in (~\<^sub>p c, Neg t'))" |
|
408 |
| "split0 (CNP n c t) = (let (c', t') = split0 t in (c', CNP n c t'))" |
|
409 |
| "split0 (Add s t) = (let (c1, s') = split0 s; (c2, t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))" |
|
410 |
| "split0 (Sub s t) = (let (c1, s') = split0 s; (c2, t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))" |
|
411 |
| "split0 (Mul c t) = (let (c', t') = split0 t in (c *\<^sub>p c', Mul c t'))" |
|
41822 | 412 |
| "split0 t = (0\<^sub>p, t)" |
413 |
||
414 |
declare let_cong[fundef_cong] |
|
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415 |
|
55754 | 416 |
lemma split0_stupid[simp]: "\<exists>x y. (x, y) = split0 p" |
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417 |
apply (rule exI[where x="fst (split0 p)"]) |
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418 |
apply (rule exI[where x="snd (split0 p)"]) |
55754 | 419 |
apply simp |
420 |
done |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
421 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
422 |
lemma split0: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
423 |
"tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
424 |
apply (induct t rule: split0.induct) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
425 |
apply simp |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
426 |
apply (simp add: Let_def split_def field_simps) |
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
427 |
apply (simp add: Let_def split_def field_simps) |
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
428 |
apply (simp add: Let_def split_def field_simps) |
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
429 |
apply (simp add: Let_def split_def field_simps) |
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
430 |
apply (simp add: Let_def split_def field_simps) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
431 |
apply (simp add: Let_def split_def mult.assoc distrib_left[symmetric]) |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
432 |
apply (simp add: Let_def split_def field_simps) |
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
433 |
apply (simp add: Let_def split_def field_simps) |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
434 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
435 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
436 |
lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')" |
55754 | 437 |
proof - |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
438 |
fix c' t' |
55754 | 439 |
assume "split0 t = (c', t')" |
55768 | 440 |
then have "c' = fst (split0 t)" and "t' = snd (split0 t)" |
55754 | 441 |
by auto |
55768 | 442 |
with split0[where t="t" and bs="bs"] |
443 |
show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" |
|
55754 | 444 |
by simp |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
445 |
qed |
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
446 |
|
55754 | 447 |
lemma split0_nb0: |
36409 | 448 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
449 |
shows "split0 t = (c',t') \<Longrightarrow> tmbound 0 t'" |
55754 | 450 |
proof - |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
451 |
fix c' t' |
55754 | 452 |
assume "split0 t = (c', t')" |
55768 | 453 |
then have "c' = fst (split0 t)" and "t' = snd (split0 t)" |
55754 | 454 |
by auto |
455 |
with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" |
|
456 |
by simp |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
457 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
458 |
|
55754 | 459 |
lemma split0_nb0'[simp]: |
460 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
461 |
shows "tmbound0 (snd (split0 t))" |
55754 | 462 |
using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] |
463 |
by (simp add: tmbound0_tmbound_iff) |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
464 |
|
55754 | 465 |
lemma split0_nb: |
466 |
assumes nb: "tmbound n t" |
|
467 |
shows "tmbound n (snd (split0 t))" |
|
468 |
using nb by (induct t rule: split0.induct) (auto simp add: Let_def split_def) |
|
33152
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chaieb
parents:
diff
changeset
|
469 |
|
55754 | 470 |
lemma split0_blt: |
471 |
assumes nb: "tmboundslt n t" |
|
472 |
shows "tmboundslt n (snd (split0 t))" |
|
473 |
using nb by (induct t rule: split0.induct) (auto simp add: Let_def split_def) |
|
33152
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chaieb
parents:
diff
changeset
|
474 |
|
55754 | 475 |
lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst (split0 t)) = 0" |
476 |
by (induct t rule: split0.induct) (auto simp add: Let_def split_def) |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
477 |
|
55754 | 478 |
lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst (split0 t)) = 0 \<or> n > 0" |
479 |
by (induct t rule: split0.induct) (auto simp add: Let_def split_def) |
|
480 |
||
481 |
lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst (split0 t)) = 0" |
|
482 |
by (induct t rule: split0.induct) (auto simp add: Let_def split_def) |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
483 |
|
241cfaed158f
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chaieb
parents:
diff
changeset
|
484 |
lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))" |
55754 | 485 |
by (induct p rule: split0.induct) (auto simp add: isnpoly_def Let_def split_def) |
33152
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chaieb
parents:
diff
changeset
|
486 |
|
55754 | 487 |
lemma isnpoly_fst_split0: |
488 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
489 |
shows "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))" |
|
490 |
by (induct p rule: split0.induct) |
|
491 |
(auto simp add: polyadd_norm polysub_norm polyneg_norm polymul_norm Let_def split_def) |
|
492 |
||
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
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parents:
diff
changeset
|
493 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
494 |
subsection{* Formulae *} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
495 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
496 |
datatype fm = T| F| Le tm | Lt tm | Eq tm | NEq tm| |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
497 |
NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
498 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
499 |
|
55754 | 500 |
(* A size for fm *) |
501 |
fun fmsize :: "fm \<Rightarrow> nat" |
|
502 |
where |
|
33152
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chaieb
parents:
diff
changeset
|
503 |
"fmsize (NOT p) = 1 + fmsize p" |
41822 | 504 |
| "fmsize (And p q) = 1 + fmsize p + fmsize q" |
505 |
| "fmsize (Or p q) = 1 + fmsize p + fmsize q" |
|
506 |
| "fmsize (Imp p q) = 3 + fmsize p + fmsize q" |
|
507 |
| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" |
|
508 |
| "fmsize (E p) = 1 + fmsize p" |
|
509 |
| "fmsize (A p) = 4+ fmsize p" |
|
510 |
| "fmsize p = 1" |
|
55754 | 511 |
|
512 |
(* several lemmas about fmsize *) |
|
513 |
lemma fmsize_pos[termination_simp]: "fmsize p > 0" |
|
514 |
by (induct p rule: fmsize.induct) simp_all |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
515 |
|
241cfaed158f
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chaieb
parents:
diff
changeset
|
516 |
(* Semantics of formulae (fm) *) |
55768 | 517 |
primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" |
518 |
where |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
519 |
"Ifm vs bs T = True" |
39246 | 520 |
| "Ifm vs bs F = False" |
521 |
| "Ifm vs bs (Lt a) = (Itm vs bs a < 0)" |
|
522 |
| "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)" |
|
523 |
| "Ifm vs bs (Eq a) = (Itm vs bs a = 0)" |
|
524 |
| "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)" |
|
525 |
| "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))" |
|
526 |
| "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)" |
|
527 |
| "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)" |
|
528 |
| "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))" |
|
529 |
| "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)" |
|
55754 | 530 |
| "Ifm vs bs (E p) = (\<exists>x. Ifm vs (x#bs) p)" |
531 |
| "Ifm vs bs (A p) = (\<forall>x. Ifm vs (x#bs) p)" |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
532 |
|
55768 | 533 |
fun not:: "fm \<Rightarrow> fm" |
534 |
where |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
535 |
"not (NOT (NOT p)) = not p" |
41822 | 536 |
| "not (NOT p) = p" |
537 |
| "not T = F" |
|
538 |
| "not F = T" |
|
539 |
| "not (Lt t) = Le (tmneg t)" |
|
540 |
| "not (Le t) = Lt (tmneg t)" |
|
541 |
| "not (Eq t) = NEq t" |
|
542 |
| "not (NEq t) = Eq t" |
|
543 |
| "not p = NOT p" |
|
55754 | 544 |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
545 |
lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)" |
55754 | 546 |
by (induct p rule: not.induct) auto |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
547 |
|
55754 | 548 |
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
549 |
where |
|
550 |
"conj p q \<equiv> |
|
551 |
(if p = F \<or> q = F then F |
|
552 |
else if p = T then q |
|
553 |
else if q = T then p |
|
554 |
else if p = q then p |
|
555 |
else And p q)" |
|
556 |
||
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
557 |
lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)" |
55754 | 558 |
by (cases "p=F \<or> q=F", simp_all add: conj_def) (cases p, simp_all) |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
559 |
|
55754 | 560 |
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
561 |
where |
|
562 |
"disj p q \<equiv> |
|
563 |
(if (p = T \<or> q = T) then T |
|
564 |
else if p = F then q |
|
565 |
else if q = F then p |
|
566 |
else if p = q then p |
|
567 |
else Or p q)" |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
568 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
569 |
lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)" |
55768 | 570 |
by (cases "p = T \<or> q = T", simp_all add: disj_def) (cases p, simp_all) |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
571 |
|
55754 | 572 |
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
573 |
where |
|
574 |
"imp p q \<equiv> |
|
575 |
(if p = F \<or> q = T \<or> p = q then T |
|
576 |
else if p = T then q |
|
577 |
else if q = F then not p |
|
578 |
else Imp p q)" |
|
579 |
||
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
580 |
lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)" |
55768 | 581 |
by (cases "p = F \<or> q = T") (simp_all add: imp_def) |
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
582 |
|
55754 | 583 |
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
584 |
where |
|
585 |
"iff p q \<equiv> |
|
586 |
(if p = q then T |
|
587 |
else if p = NOT q \<or> NOT p = q then F |
|
588 |
else if p = F then not q |
|
589 |
else if q = F then not p |
|
590 |
else if p = T then q |
|
591 |
else if q = T then p |
|
592 |
else Iff p q)" |
|
593 |
||
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
594 |
lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)" |
55768 | 595 |
by (unfold iff_def, cases "p = q", simp, cases "p = NOT q", simp) (cases "NOT p= q", auto) |
41822 | 596 |
|
55754 | 597 |
(* Quantifier freeness *) |
598 |
fun qfree:: "fm \<Rightarrow> bool" |
|
599 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
600 |
"qfree (E p) = False" |
41822 | 601 |
| "qfree (A p) = False" |
55754 | 602 |
| "qfree (NOT p) = qfree p" |
603 |
| "qfree (And p q) = (qfree p \<and> qfree q)" |
|
604 |
| "qfree (Or p q) = (qfree p \<and> qfree q)" |
|
605 |
| "qfree (Imp p q) = (qfree p \<and> qfree q)" |
|
41822 | 606 |
| "qfree (Iff p q) = (qfree p \<and> qfree q)" |
607 |
| "qfree p = True" |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
608 |
|
55754 | 609 |
(* Boundedness and substitution *) |
610 |
primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" |
|
611 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
612 |
"boundslt n T = True" |
39246 | 613 |
| "boundslt n F = True" |
55768 | 614 |
| "boundslt n (Lt t) = tmboundslt n t" |
615 |
| "boundslt n (Le t) = tmboundslt n t" |
|
616 |
| "boundslt n (Eq t) = tmboundslt n t" |
|
617 |
| "boundslt n (NEq t) = tmboundslt n t" |
|
39246 | 618 |
| "boundslt n (NOT p) = boundslt n p" |
619 |
| "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)" |
|
620 |
| "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)" |
|
621 |
| "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))" |
|
622 |
| "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)" |
|
623 |
| "boundslt n (E p) = boundslt (Suc n) p" |
|
624 |
| "boundslt n (A p) = boundslt (Suc n) p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
625 |
|
55754 | 626 |
fun bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) |
627 |
where |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
628 |
"bound0 T = True" |
41822 | 629 |
| "bound0 F = True" |
630 |
| "bound0 (Lt a) = tmbound0 a" |
|
631 |
| "bound0 (Le a) = tmbound0 a" |
|
632 |
| "bound0 (Eq a) = tmbound0 a" |
|
633 |
| "bound0 (NEq a) = tmbound0 a" |
|
634 |
| "bound0 (NOT p) = bound0 p" |
|
635 |
| "bound0 (And p q) = (bound0 p \<and> bound0 q)" |
|
636 |
| "bound0 (Or p q) = (bound0 p \<and> bound0 q)" |
|
637 |
| "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))" |
|
638 |
| "bound0 (Iff p q) = (bound0 p \<and> bound0 q)" |
|
639 |
| "bound0 p = False" |
|
55754 | 640 |
|
33152
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Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
641 |
lemma bound0_I: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
642 |
assumes bp: "bound0 p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
643 |
shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p" |
55754 | 644 |
using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"] |
645 |
by (induct p rule: bound0.induct) auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
646 |
|
55754 | 647 |
primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) |
648 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
649 |
"bound m T = True" |
39246 | 650 |
| "bound m F = True" |
651 |
| "bound m (Lt t) = tmbound m t" |
|
652 |
| "bound m (Le t) = tmbound m t" |
|
653 |
| "bound m (Eq t) = tmbound m t" |
|
654 |
| "bound m (NEq t) = tmbound m t" |
|
655 |
| "bound m (NOT p) = bound m p" |
|
656 |
| "bound m (And p q) = (bound m p \<and> bound m q)" |
|
657 |
| "bound m (Or p q) = (bound m p \<and> bound m q)" |
|
658 |
| "bound m (Imp p q) = ((bound m p) \<and> (bound m q))" |
|
659 |
| "bound m (Iff p q) = (bound m p \<and> bound m q)" |
|
660 |
| "bound m (E p) = bound (Suc m) p" |
|
661 |
| "bound m (A p) = bound (Suc m) p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
662 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
663 |
lemma bound_I: |
55754 | 664 |
assumes bnd: "boundslt (length bs) p" |
665 |
and nb: "bound n p" |
|
666 |
and le: "n \<le> length bs" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
667 |
shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
668 |
using bnd nb le tmbound_I[where bs=bs and vs = vs] |
55754 | 669 |
proof (induct p arbitrary: bs n rule: fm.induct) |
670 |
case (E p bs n) |
|
671 |
{ |
|
672 |
fix y |
|
673 |
from E have bnd: "boundslt (length (y#bs)) p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
674 |
and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+ |
55754 | 675 |
from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" . |
676 |
} |
|
55768 | 677 |
then show ?case by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
678 |
next |
55754 | 679 |
case (A p bs n) |
680 |
{ |
|
681 |
fix y |
|
682 |
from A have bnd: "boundslt (length (y#bs)) p" |
|
683 |
and nb: "bound (Suc n) p" |
|
684 |
and le: "Suc n \<le> length (y#bs)" |
|
685 |
by simp_all |
|
686 |
from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" . |
|
687 |
} |
|
55768 | 688 |
then show ?case by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
689 |
qed auto |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
690 |
|
55768 | 691 |
fun decr0 :: "fm \<Rightarrow> fm" |
692 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
693 |
"decr0 (Lt a) = Lt (decrtm0 a)" |
41822 | 694 |
| "decr0 (Le a) = Le (decrtm0 a)" |
695 |
| "decr0 (Eq a) = Eq (decrtm0 a)" |
|
696 |
| "decr0 (NEq a) = NEq (decrtm0 a)" |
|
55754 | 697 |
| "decr0 (NOT p) = NOT (decr0 p)" |
41822 | 698 |
| "decr0 (And p q) = conj (decr0 p) (decr0 q)" |
699 |
| "decr0 (Or p q) = disj (decr0 p) (decr0 q)" |
|
700 |
| "decr0 (Imp p q) = imp (decr0 p) (decr0 q)" |
|
701 |
| "decr0 (Iff p q) = iff (decr0 p) (decr0 q)" |
|
702 |
| "decr0 p = p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
703 |
|
55754 | 704 |
lemma decr0: |
705 |
assumes nb: "bound0 p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
706 |
shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)" |
55754 | 707 |
using nb |
708 |
by (induct p rule: decr0.induct) (simp_all add: decrtm0) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
709 |
|
55754 | 710 |
primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" |
711 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
712 |
"decr m T = T" |
39246 | 713 |
| "decr m F = F" |
714 |
| "decr m (Lt t) = (Lt (decrtm m t))" |
|
715 |
| "decr m (Le t) = (Le (decrtm m t))" |
|
716 |
| "decr m (Eq t) = (Eq (decrtm m t))" |
|
717 |
| "decr m (NEq t) = (NEq (decrtm m t))" |
|
55754 | 718 |
| "decr m (NOT p) = NOT (decr m p)" |
39246 | 719 |
| "decr m (And p q) = conj (decr m p) (decr m q)" |
720 |
| "decr m (Or p q) = disj (decr m p) (decr m q)" |
|
721 |
| "decr m (Imp p q) = imp (decr m p) (decr m q)" |
|
722 |
| "decr m (Iff p q) = iff (decr m p) (decr m q)" |
|
723 |
| "decr m (E p) = E (decr (Suc m) p)" |
|
724 |
| "decr m (A p) = A (decr (Suc m) p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
725 |
|
55754 | 726 |
lemma decr: |
727 |
assumes bnd: "boundslt (length bs) p" |
|
728 |
and nb: "bound m p" |
|
729 |
and nle: "m < length bs" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
730 |
shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
731 |
using bnd nb nle |
55754 | 732 |
proof (induct p arbitrary: bs m rule: fm.induct) |
733 |
case (E p bs m) |
|
734 |
{ fix x |
|
735 |
from E |
|
736 |
have bnd: "boundslt (length (x#bs)) p" |
|
737 |
and nb: "bound (Suc m) p" |
|
738 |
and nle: "Suc m < length (x#bs)" |
|
739 |
by auto |
|
740 |
from E(1)[OF bnd nb nle] |
|
741 |
have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" . |
|
742 |
} |
|
55768 | 743 |
then show ?case by auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
744 |
next |
55754 | 745 |
case (A p bs m) |
746 |
{ fix x |
|
747 |
from A |
|
748 |
have bnd: "boundslt (length (x#bs)) p" |
|
749 |
and nb: "bound (Suc m) p" |
|
750 |
and nle: "Suc m < length (x#bs)" |
|
751 |
by auto |
|
752 |
from A(1)[OF bnd nb nle] |
|
753 |
have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" . |
|
754 |
} |
|
55768 | 755 |
then show ?case by auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
756 |
qed (auto simp add: decrtm removen_nth) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
757 |
|
55754 | 758 |
primrec subst0 :: "tm \<Rightarrow> fm \<Rightarrow> fm" |
759 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
760 |
"subst0 t T = T" |
39246 | 761 |
| "subst0 t F = F" |
762 |
| "subst0 t (Lt a) = Lt (tmsubst0 t a)" |
|
763 |
| "subst0 t (Le a) = Le (tmsubst0 t a)" |
|
764 |
| "subst0 t (Eq a) = Eq (tmsubst0 t a)" |
|
765 |
| "subst0 t (NEq a) = NEq (tmsubst0 t a)" |
|
766 |
| "subst0 t (NOT p) = NOT (subst0 t p)" |
|
767 |
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" |
|
768 |
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" |
|
769 |
| "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" |
|
770 |
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" |
|
771 |
| "subst0 t (E p) = E p" |
|
772 |
| "subst0 t (A p) = A p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
773 |
|
55754 | 774 |
lemma subst0: |
775 |
assumes qf: "qfree p" |
|
776 |
shows "Ifm vs (x # bs) (subst0 t p) = Ifm vs ((Itm vs (x # bs) t) # bs) p" |
|
777 |
using qf tmsubst0[where x="x" and bs="bs" and t="t"] |
|
778 |
by (induct p rule: fm.induct) auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
779 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
780 |
lemma subst0_nb: |
55754 | 781 |
assumes bp: "tmbound0 t" |
782 |
and qf: "qfree p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
783 |
shows "bound0 (subst0 t p)" |
55754 | 784 |
using qf tmsubst0_nb[OF bp] bp |
785 |
by (induct p rule: fm.induct) auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
786 |
|
55754 | 787 |
primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" |
788 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
789 |
"subst n t T = T" |
39246 | 790 |
| "subst n t F = F" |
791 |
| "subst n t (Lt a) = Lt (tmsubst n t a)" |
|
792 |
| "subst n t (Le a) = Le (tmsubst n t a)" |
|
793 |
| "subst n t (Eq a) = Eq (tmsubst n t a)" |
|
794 |
| "subst n t (NEq a) = NEq (tmsubst n t a)" |
|
795 |
| "subst n t (NOT p) = NOT (subst n t p)" |
|
796 |
| "subst n t (And p q) = And (subst n t p) (subst n t q)" |
|
797 |
| "subst n t (Or p q) = Or (subst n t p) (subst n t q)" |
|
798 |
| "subst n t (Imp p q) = Imp (subst n t p) (subst n t q)" |
|
799 |
| "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)" |
|
800 |
| "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)" |
|
801 |
| "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
802 |
|
55754 | 803 |
lemma subst: |
804 |
assumes nb: "boundslt (length bs) p" |
|
805 |
and nlm: "n \<le> length bs" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
806 |
shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
807 |
using nb nlm |
39246 | 808 |
proof (induct p arbitrary: bs n t rule: fm.induct) |
55754 | 809 |
case (E p bs n) |
810 |
{ |
|
811 |
fix x |
|
812 |
from E have bn: "boundslt (length (x#bs)) p" |
|
813 |
by simp |
|
814 |
from E have nlm: "Suc n \<le> length (x#bs)" |
|
815 |
by simp |
|
816 |
from E(1)[OF bn nlm] |
|
55768 | 817 |
have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = |
818 |
Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" |
|
55754 | 819 |
by simp |
55768 | 820 |
then have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = |
821 |
Ifm vs (x#bs[n:= Itm vs bs t]) p" |
|
55754 | 822 |
by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) |
823 |
} |
|
55768 | 824 |
then show ?case by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
825 |
next |
55754 | 826 |
case (A p bs n) |
827 |
{ |
|
828 |
fix x |
|
829 |
from A have bn: "boundslt (length (x#bs)) p" |
|
830 |
by simp |
|
831 |
from A have nlm: "Suc n \<le> length (x#bs)" |
|
832 |
by simp |
|
833 |
from A(1)[OF bn nlm] |
|
55768 | 834 |
have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = |
835 |
Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" |
|
55754 | 836 |
by simp |
55768 | 837 |
then have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = |
838 |
Ifm vs (x#bs[n:= Itm vs bs t]) p" |
|
55754 | 839 |
by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) |
840 |
} |
|
55768 | 841 |
then show ?case by simp |
55754 | 842 |
qed (auto simp add: tmsubst) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
843 |
|
55754 | 844 |
lemma subst_nb: |
845 |
assumes tnb: "tmbound m t" |
|
846 |
shows "bound m (subst m t p)" |
|
847 |
using tnb tmsubst_nb incrtm0_tmbound |
|
848 |
by (induct p arbitrary: m t rule: fm.induct) auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
849 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
850 |
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)" |
55754 | 851 |
by (induct p rule: not.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
852 |
lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)" |
55754 | 853 |
by (induct p rule: not.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
854 |
lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)" |
55754 | 855 |
by (induct p rule: not.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
856 |
lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)" |
55754 | 857 |
by (induct p rule: not.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
858 |
|
55754 | 859 |
lemma conj_qf[simp]: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (conj p q)" |
860 |
using conj_def by auto |
|
861 |
lemma conj_nb0[simp]: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (conj p q)" |
|
862 |
using conj_def by auto |
|
863 |
lemma conj_nb[simp]: "bound n p \<Longrightarrow> bound n q \<Longrightarrow> bound n (conj p q)" |
|
864 |
using conj_def by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
865 |
lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)" |
55754 | 866 |
using conj_def by auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
867 |
|
55754 | 868 |
lemma disj_qf[simp]: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (disj p q)" |
869 |
using disj_def by auto |
|
870 |
lemma disj_nb0[simp]: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (disj p q)" |
|
871 |
using disj_def by auto |
|
872 |
lemma disj_nb[simp]: "bound n p \<Longrightarrow> bound n q \<Longrightarrow> bound n (disj p q)" |
|
873 |
using disj_def by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
874 |
lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)" |
55754 | 875 |
using disj_def by auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
876 |
|
55754 | 877 |
lemma imp_qf[simp]: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (imp p q)" |
55768 | 878 |
using imp_def by (cases "p = F \<or> q = T") (simp_all add: imp_def) |
55754 | 879 |
lemma imp_nb0[simp]: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (imp p q)" |
55768 | 880 |
using imp_def by (cases "p = F \<or> q = T \<or> p = q") (simp_all add: imp_def) |
55754 | 881 |
lemma imp_nb[simp]: "bound n p \<Longrightarrow> bound n q \<Longrightarrow> bound n (imp p q)" |
55768 | 882 |
using imp_def by (cases "p = F \<or> q = T \<or> p = q") (simp_all add: imp_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
883 |
lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)" |
55754 | 884 |
using imp_def by auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
885 |
|
55754 | 886 |
lemma iff_qf[simp]: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (iff p q)" |
887 |
unfolding iff_def by (cases "p = q") auto |
|
888 |
lemma iff_nb0[simp]: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (iff p q)" |
|
889 |
using iff_def unfolding iff_def by (cases "p = q") auto |
|
890 |
lemma iff_nb[simp]: "bound n p \<Longrightarrow> bound n q \<Longrightarrow> bound n (iff p q)" |
|
891 |
using iff_def unfolding iff_def by (cases "p = q") auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
892 |
lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)" |
55754 | 893 |
using iff_def by auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
894 |
lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)" |
55754 | 895 |
by (induct p) simp_all |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
896 |
|
55754 | 897 |
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) |
898 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
899 |
"isatom T = True" |
41822 | 900 |
| "isatom F = True" |
901 |
| "isatom (Lt a) = True" |
|
902 |
| "isatom (Le a) = True" |
|
903 |
| "isatom (Eq a) = True" |
|
904 |
| "isatom (NEq a) = True" |
|
905 |
| "isatom p = False" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
906 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
907 |
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" |
55754 | 908 |
by (induct p) simp_all |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
909 |
|
55754 | 910 |
definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" |
911 |
where |
|
912 |
"djf f p q \<equiv> |
|
913 |
(if q = T then T |
|
914 |
else if q = F then f p |
|
915 |
else (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))" |
|
916 |
||
917 |
definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" |
|
918 |
where "evaldjf f ps \<equiv> foldr (djf f) ps F" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
919 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
920 |
lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)" |
55768 | 921 |
by (cases "q=T", simp add: djf_def,cases "q=F", simp add: djf_def) |
55754 | 922 |
(cases "f p", simp_all add: Let_def djf_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
923 |
|
55754 | 924 |
lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) \<longleftrightarrow> (\<exists>p \<in> set ps. Ifm vs bs (f p))" |
925 |
by (induct ps) (simp_all add: evaldjf_def djf_Or) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
926 |
|
55754 | 927 |
lemma evaldjf_bound0: |
928 |
assumes nb: "\<forall>x\<in> set xs. bound0 (f x)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
929 |
shows "bound0 (evaldjf f xs)" |
55754 | 930 |
using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
931 |
|
55754 | 932 |
lemma evaldjf_qf: |
933 |
assumes nb: "\<forall>x\<in> set xs. qfree (f x)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
934 |
shows "qfree (evaldjf f xs)" |
55754 | 935 |
using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
936 |
|
55754 | 937 |
fun disjuncts :: "fm \<Rightarrow> fm list" |
938 |
where |
|
939 |
"disjuncts (Or p q) = disjuncts p @ disjuncts q" |
|
41822 | 940 |
| "disjuncts F = []" |
941 |
| "disjuncts p = [p]" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
942 |
|
55754 | 943 |
lemma disjuncts: "(\<exists>q \<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p" |
944 |
by (induct p rule: disjuncts.induct) auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
945 |
|
55754 | 946 |
lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall>q \<in> set (disjuncts p). bound0 q" |
947 |
proof - |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
948 |
assume nb: "bound0 p" |
55768 | 949 |
then have "list_all bound0 (disjuncts p)" |
950 |
by (induct p rule:disjuncts.induct) auto |
|
951 |
then show ?thesis |
|
952 |
by (simp only: list_all_iff) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
953 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
954 |
|
55754 | 955 |
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall>q\<in> set (disjuncts p). qfree q" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
956 |
proof- |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
957 |
assume qf: "qfree p" |
55768 | 958 |
then have "list_all qfree (disjuncts p)" |
959 |
by (induct p rule: disjuncts.induct) auto |
|
960 |
then show ?thesis by (simp only: list_all_iff) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
961 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
962 |
|
55768 | 963 |
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" |
964 |
where "DJ f p \<equiv> evaldjf f (disjuncts p)" |
|
965 |
||
966 |
lemma DJ: |
|
967 |
assumes fdj: "\<forall>p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))" |
|
968 |
and fF: "f F = F" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
969 |
shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)" |
55768 | 970 |
proof - |
55754 | 971 |
have "Ifm vs bs (DJ f p) = (\<exists>q \<in> set (disjuncts p). Ifm vs bs (f q))" |
972 |
by (simp add: DJ_def evaldjf_ex) |
|
55768 | 973 |
also have "\<dots> = Ifm vs bs (f p)" |
974 |
using fdj fF by (induct p rule: disjuncts.induct) auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
975 |
finally show ?thesis . |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
976 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
977 |
|
55768 | 978 |
lemma DJ_qf: |
979 |
assumes fqf: "\<forall>p. qfree p \<longrightarrow> qfree (f p)" |
|
980 |
shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p)" |
|
981 |
proof clarify |
|
982 |
fix p |
|
983 |
assume qf: "qfree p" |
|
984 |
have th: "DJ f p = evaldjf f (disjuncts p)" |
|
985 |
by (simp add: DJ_def) |
|
55754 | 986 |
from disjuncts_qf[OF qf] have "\<forall>q\<in> set (disjuncts p). qfree q" . |
55768 | 987 |
with fqf have th':"\<forall>q\<in> set (disjuncts p). qfree (f q)" |
988 |
by blast |
|
989 |
from evaldjf_qf[OF th'] th show "qfree (DJ f p)" |
|
990 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
991 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
992 |
|
55768 | 993 |
lemma DJ_qe: |
994 |
assumes qe: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))" |
|
55754 | 995 |
shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))" |
55768 | 996 |
proof clarify |
997 |
fix p :: fm and bs |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
998 |
assume qf: "qfree p" |
55768 | 999 |
from qe have qth: "\<forall>p. qfree p \<longrightarrow> qfree (qe p)" |
1000 |
by blast |
|
1001 |
from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" |
|
1002 |
by auto |
|
1003 |
have "Ifm vs bs (DJ qe p) \<longleftrightarrow> (\<exists>q\<in> set (disjuncts p). Ifm vs bs (qe q))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1004 |
by (simp add: DJ_def evaldjf_ex) |
55768 | 1005 |
also have "\<dots> = (\<exists>q \<in> set(disjuncts p). Ifm vs bs (E q))" |
1006 |
using qe disjuncts_qf[OF qf] by auto |
|
1007 |
also have "\<dots> = Ifm vs bs (E p)" |
|
1008 |
by (induct p rule: disjuncts.induct) auto |
|
1009 |
finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" |
|
1010 |
using qfth by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1011 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1012 |
|
55768 | 1013 |
fun conjuncts :: "fm \<Rightarrow> fm list" |
1014 |
where |
|
1015 |
"conjuncts (And p q) = conjuncts p @ conjuncts q" |
|
41822 | 1016 |
| "conjuncts T = []" |
1017 |
| "conjuncts p = [p]" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1018 |
|
55768 | 1019 |
definition list_conj :: "fm list \<Rightarrow> fm" |
1020 |
where "list_conj ps \<equiv> foldr conj ps T" |
|
1021 |
||
1022 |
definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" |
|
1023 |
where |
|
1024 |
"CJNB f p \<equiv> |
|
1025 |
(let cjs = conjuncts p; |
|
1026 |
(yes, no) = partition bound0 cjs |
|
1027 |
in conj (decr0 (list_conj yes)) (f (list_conj no)))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1028 |
|
55754 | 1029 |
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall>q\<in> set (conjuncts p). qfree q" |
55768 | 1030 |
proof - |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1031 |
assume qf: "qfree p" |
55768 | 1032 |
then have "list_all qfree (conjuncts p)" |
1033 |
by (induct p rule: conjuncts.induct) auto |
|
1034 |
then show ?thesis |
|
1035 |
by (simp only: list_all_iff) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1036 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1037 |
|
55754 | 1038 |
lemma conjuncts: "(\<forall>q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p" |
55768 | 1039 |
by (induct p rule: conjuncts.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1040 |
|
55754 | 1041 |
lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall>q\<in> set (conjuncts p). bound0 q" |
55768 | 1042 |
proof - |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1043 |
assume nb: "bound0 p" |
55768 | 1044 |
then have "list_all bound0 (conjuncts p)" |
1045 |
by (induct p rule:conjuncts.induct) auto |
|
1046 |
then show ?thesis |
|
1047 |
by (simp only: list_all_iff) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1048 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1049 |
|
55768 | 1050 |
fun islin :: "fm \<Rightarrow> bool" |
1051 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1052 |
"islin (And p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1053 |
| "islin (Or p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1054 |
| "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1055 |
| "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1056 |
| "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1057 |
| "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1058 |
| "islin (NOT p) = False" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1059 |
| "islin (Imp p q) = False" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1060 |
| "islin (Iff p q) = False" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1061 |
| "islin p = bound0 p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1062 |
|
55768 | 1063 |
lemma islin_stupid: |
1064 |
assumes nb: "tmbound0 p" |
|
1065 |
shows "islin (Lt p)" |
|
1066 |
and "islin (Le p)" |
|
1067 |
and "islin (Eq p)" |
|
1068 |
and "islin (NEq p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1069 |
using nb by (cases p, auto, case_tac nat, auto)+ |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1070 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1071 |
definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1072 |
definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)" |
55768 | 1073 |
definition "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1074 |
definition "neq p = not (eq p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1075 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1076 |
lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)" |
55768 | 1077 |
apply (simp add: lt_def) |
1078 |
apply (cases p) |
|
1079 |
apply simp_all |
|
1080 |
apply (case_tac poly) |
|
1081 |
apply (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1082 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1083 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1084 |
lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)" |
55768 | 1085 |
apply (simp add: le_def) |
1086 |
apply (cases p) |
|
1087 |
apply simp_all |
|
1088 |
apply (case_tac poly) |
|
1089 |
apply (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1090 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1091 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1092 |
lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)" |
55768 | 1093 |
apply (simp add: eq_def) |
1094 |
apply (cases p) |
|
1095 |
apply simp_all |
|
1096 |
apply (case_tac poly) |
|
1097 |
apply (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1098 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1099 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1100 |
lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)" |
55768 | 1101 |
by (simp add: neq_def eq) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1102 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1103 |
lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1104 |
apply (simp add: lt_def) |
55768 | 1105 |
apply (cases p) |
1106 |
apply simp_all |
|
1107 |
apply (case_tac poly) |
|
1108 |
apply simp_all |
|
1109 |
apply (case_tac nat) |
|
1110 |
apply simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1111 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1112 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1113 |
lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1114 |
apply (simp add: le_def) |
55768 | 1115 |
apply (cases p) |
1116 |
apply simp_all |
|
1117 |
apply (case_tac poly) |
|
1118 |
apply simp_all |
|
1119 |
apply (case_tac nat) |
|
1120 |
apply simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1121 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1122 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1123 |
lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1124 |
apply (simp add: eq_def) |
55768 | 1125 |
apply (cases p) |
1126 |
apply simp_all |
|
1127 |
apply (case_tac poly) |
|
1128 |
apply simp_all |
|
1129 |
apply (case_tac nat) |
|
1130 |
apply simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1131 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1132 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1133 |
lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1134 |
apply (simp add: neq_def eq_def) |
55768 | 1135 |
apply (cases p) |
1136 |
apply simp_all |
|
1137 |
apply (case_tac poly) |
|
1138 |
apply simp_all |
|
1139 |
apply (case_tac nat) |
|
1140 |
apply simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1141 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1142 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1143 |
definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1144 |
definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1145 |
definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1146 |
definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1147 |
|
55768 | 1148 |
lemma simplt_islin[simp]: |
1149 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1150 |
shows "islin (simplt t)" |
55754 | 1151 |
unfolding simplt_def |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1152 |
using split0_nb0' |
55768 | 1153 |
by (auto simp add: lt_lin Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] |
1154 |
islin_stupid allpolys_split0[OF simptm_allpolys_npoly]) |
|
1155 |
||
1156 |
lemma simple_islin[simp]: |
|
1157 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1158 |
shows "islin (simple t)" |
55754 | 1159 |
unfolding simple_def |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1160 |
using split0_nb0' |
55768 | 1161 |
by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] |
1162 |
islin_stupid allpolys_split0[OF simptm_allpolys_npoly] le_lin) |
|
1163 |
||
1164 |
lemma simpeq_islin[simp]: |
|
1165 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1166 |
shows "islin (simpeq t)" |
55754 | 1167 |
unfolding simpeq_def |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1168 |
using split0_nb0' |
55768 | 1169 |
by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] |
1170 |
islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin) |
|
1171 |
||
1172 |
lemma simpneq_islin[simp]: |
|
1173 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1174 |
shows "islin (simpneq t)" |
55754 | 1175 |
unfolding simpneq_def |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1176 |
using split0_nb0' |
55768 | 1177 |
by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] |
1178 |
islin_stupid allpolys_split0[OF simptm_allpolys_npoly] neq_lin) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1179 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1180 |
lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)" |
55768 | 1181 |
by (cases "split0 s") auto |
1182 |
||
1183 |
lemma split0_npoly: |
|
1184 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
1185 |
and n: "allpolys isnpoly t" |
|
1186 |
shows "isnpoly (fst (split0 t))" |
|
1187 |
and "allpolys isnpoly (snd (split0 t))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1188 |
using n |
55768 | 1189 |
by (induct t rule: split0.induct) |
1190 |
(auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm |
|
1191 |
polysub_norm really_stupid) |
|
1192 |
||
1193 |
lemma simplt[simp]: "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)" |
|
1194 |
proof - |
|
1195 |
have n: "allpolys isnpoly (simptm t)" |
|
1196 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1197 |
let ?t = "simptm t" |
55768 | 1198 |
{ |
1199 |
assume "fst (split0 ?t) = 0\<^sub>p" |
|
1200 |
then have ?thesis |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1201 |
using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs] |
55768 | 1202 |
by (simp add: simplt_def Let_def split_def lt) |
1203 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1204 |
moreover |
55768 | 1205 |
{ |
1206 |
assume "fst (split0 ?t) \<noteq> 0\<^sub>p" |
|
1207 |
then have ?thesis |
|
1208 |
using split0[of "simptm t" vs bs] |
|
1209 |
by (simp add: simplt_def Let_def split_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1210 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1211 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1212 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1213 |
|
55768 | 1214 |
lemma simple[simp]: "Ifm vs bs (simple t) = Ifm vs bs (Le t)" |
1215 |
proof - |
|
1216 |
have n: "allpolys isnpoly (simptm t)" |
|
1217 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1218 |
let ?t = "simptm t" |
55768 | 1219 |
{ |
1220 |
assume "fst (split0 ?t) = 0\<^sub>p" |
|
1221 |
then have ?thesis |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1222 |
using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs] |
55768 | 1223 |
by (simp add: simple_def Let_def split_def le) |
1224 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1225 |
moreover |
55768 | 1226 |
{ |
1227 |
assume "fst (split0 ?t) \<noteq> 0\<^sub>p" |
|
1228 |
then have ?thesis |
|
1229 |
using split0[of "simptm t" vs bs] |
|
1230 |
by (simp add: simple_def Let_def split_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1231 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1232 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1233 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1234 |
|
55768 | 1235 |
lemma simpeq[simp]: "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)" |
1236 |
proof - |
|
1237 |
have n: "allpolys isnpoly (simptm t)" |
|
1238 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1239 |
let ?t = "simptm t" |
55768 | 1240 |
{ |
1241 |
assume "fst (split0 ?t) = 0\<^sub>p" |
|
1242 |
then have ?thesis |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1243 |
using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs] |
55768 | 1244 |
by (simp add: simpeq_def Let_def split_def) |
1245 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1246 |
moreover |
55768 | 1247 |
{ |
1248 |
assume "fst (split0 ?t) \<noteq> 0\<^sub>p" |
|
1249 |
then have ?thesis using split0[of "simptm t" vs bs] |
|
1250 |
by (simp add: simpeq_def Let_def split_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1251 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1252 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1253 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1254 |
|
55768 | 1255 |
lemma simpneq[simp]: "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)" |
1256 |
proof - |
|
1257 |
have n: "allpolys isnpoly (simptm t)" |
|
1258 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1259 |
let ?t = "simptm t" |
55768 | 1260 |
{ |
1261 |
assume "fst (split0 ?t) = 0\<^sub>p" |
|
1262 |
then have ?thesis |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1263 |
using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs] |
55768 | 1264 |
by (simp add: simpneq_def Let_def split_def) |
1265 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1266 |
moreover |
55768 | 1267 |
{ |
1268 |
assume "fst (split0 ?t) \<noteq> 0\<^sub>p" |
|
1269 |
then have ?thesis |
|
1270 |
using split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1271 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1272 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1273 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1274 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1275 |
lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1276 |
apply (simp add: lt_def) |
55768 | 1277 |
apply (cases t) |
1278 |
apply auto |
|
1279 |
apply (case_tac poly) |
|
1280 |
apply auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1281 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1282 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1283 |
lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1284 |
apply (simp add: le_def) |
55768 | 1285 |
apply (cases t) |
1286 |
apply auto |
|
1287 |
apply (case_tac poly) |
|
1288 |
apply auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1289 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1290 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1291 |
lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1292 |
apply (simp add: eq_def) |
55768 | 1293 |
apply (cases t) |
1294 |
apply auto |
|
1295 |
apply (case_tac poly) |
|
1296 |
apply auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1297 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1298 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1299 |
lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1300 |
apply (simp add: neq_def eq_def) |
55768 | 1301 |
apply (cases t) |
1302 |
apply auto |
|
1303 |
apply (case_tac poly) |
|
1304 |
apply auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1305 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1306 |
|
55768 | 1307 |
lemma simplt_nb[simp]: |
1308 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1309 |
shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)" |
55768 | 1310 |
proof (simp add: simplt_def Let_def split_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1311 |
assume nb: "tmbound0 t" |
55768 | 1312 |
then have nb': "tmbound0 (simptm t)" |
1313 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1314 |
let ?c = "fst (split0 (simptm t))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1315 |
from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] |
55768 | 1316 |
have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" |
1317 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1318 |
from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] |
55768 | 1319 |
have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" |
1320 |
by (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1321 |
from iffD1[OF isnpolyh_unique[OF ths] th] |
55754 | 1322 |
have "fst (split0 (simptm t)) = 0\<^sub>p" . |
55768 | 1323 |
then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and> |
1324 |
fst (split0 (simptm t)) = 0\<^sub>p" |
|
1325 |
by (simp add: simplt_def Let_def split_def lt_nb) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1326 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1327 |
|
55768 | 1328 |
lemma simple_nb[simp]: |
1329 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1330 |
shows "tmbound0 t \<Longrightarrow> bound0 (simple t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1331 |
proof(simp add: simple_def Let_def split_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1332 |
assume nb: "tmbound0 t" |
55768 | 1333 |
then have nb': "tmbound0 (simptm t)" |
1334 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1335 |
let ?c = "fst (split0 (simptm t))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1336 |
from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] |
55768 | 1337 |
have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" |
1338 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1339 |
from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] |
55768 | 1340 |
have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" |
1341 |
by (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1342 |
from iffD1[OF isnpolyh_unique[OF ths] th] |
55754 | 1343 |
have "fst (split0 (simptm t)) = 0\<^sub>p" . |
55768 | 1344 |
then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and> |
1345 |
fst (split0 (simptm t)) = 0\<^sub>p" |
|
1346 |
by (simp add: simplt_def Let_def split_def le_nb) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1347 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1348 |
|
55768 | 1349 |
lemma simpeq_nb[simp]: |
1350 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1351 |
shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)" |
55768 | 1352 |
proof (simp add: simpeq_def Let_def split_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1353 |
assume nb: "tmbound0 t" |
55768 | 1354 |
then have nb': "tmbound0 (simptm t)" |
1355 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1356 |
let ?c = "fst (split0 (simptm t))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1357 |
from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] |
55768 | 1358 |
have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" |
1359 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1360 |
from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] |
55768 | 1361 |
have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" |
1362 |
by (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1363 |
from iffD1[OF isnpolyh_unique[OF ths] th] |
55754 | 1364 |
have "fst (split0 (simptm t)) = 0\<^sub>p" . |
55768 | 1365 |
then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and> |
1366 |
fst (split0 (simptm t)) = 0\<^sub>p" |
|
1367 |
by (simp add: simpeq_def Let_def split_def eq_nb) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1368 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1369 |
|
55768 | 1370 |
lemma simpneq_nb[simp]: |
1371 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1372 |
shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)" |
55768 | 1373 |
proof (simp add: simpneq_def Let_def split_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1374 |
assume nb: "tmbound0 t" |
55768 | 1375 |
then have nb': "tmbound0 (simptm t)" |
1376 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1377 |
let ?c = "fst (split0 (simptm t))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1378 |
from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] |
55768 | 1379 |
have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" |
1380 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1381 |
from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] |
55768 | 1382 |
have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" |
1383 |
by (simp_all add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1384 |
from iffD1[OF isnpolyh_unique[OF ths] th] |
55754 | 1385 |
have "fst (split0 (simptm t)) = 0\<^sub>p" . |
55768 | 1386 |
then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and> |
1387 |
fst (split0 (simptm t)) = 0\<^sub>p" |
|
1388 |
by (simp add: simpneq_def Let_def split_def neq_nb) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1389 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1390 |
|
55768 | 1391 |
fun conjs :: "fm \<Rightarrow> fm list" |
1392 |
where |
|
1393 |
"conjs (And p q) = conjs p @ conjs q" |
|
41822 | 1394 |
| "conjs T = []" |
1395 |
| "conjs p = [p]" |
|
55768 | 1396 |
|
55754 | 1397 |
lemma conjs_ci: "(\<forall>q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p" |
55768 | 1398 |
by (induct p rule: conjs.induct) auto |
1399 |
||
1400 |
definition list_disj :: "fm list \<Rightarrow> fm" |
|
1401 |
where "list_disj ps \<equiv> foldr disj ps F" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1402 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1403 |
lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)" |
55768 | 1404 |
by (induct ps) (auto simp add: list_conj_def) |
1405 |
||
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1406 |
lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)" |
55768 | 1407 |
by (induct ps) (auto simp add: list_conj_def) |
1408 |
||
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1409 |
lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)" |
55768 | 1410 |
by (induct ps) (auto simp add: list_disj_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1411 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1412 |
lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1413 |
unfolding conj_def by auto |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1414 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1415 |
lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q" |
55754 | 1416 |
apply (induct p rule: conjs.induct) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1417 |
apply (unfold conjs.simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1418 |
apply (unfold set_append) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1419 |
apply (unfold ball_Un) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1420 |
apply (unfold bound.simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1421 |
apply auto |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1422 |
done |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1423 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1424 |
lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q" |
55754 | 1425 |
apply (induct p rule: conjs.induct) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1426 |
apply (unfold conjs.simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1427 |
apply (unfold set_append) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1428 |
apply (unfold ball_Un) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1429 |
apply (unfold boundslt.simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1430 |
apply blast |
55768 | 1431 |
apply simp_all |
1432 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1433 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1434 |
lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1435 |
unfolding list_conj_def |
55768 | 1436 |
by (induct ps) auto |
1437 |
||
1438 |
lemma list_conj_nb: |
|
1439 |
assumes bnd: "\<forall>p\<in> set ps. bound n p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1440 |
shows "bound n (list_conj ps)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1441 |
using bnd |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1442 |
unfolding list_conj_def |
55768 | 1443 |
by (induct ps) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1444 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1445 |
lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)" |
55768 | 1446 |
unfolding list_conj_def by (induct ps) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1447 |
|
55754 | 1448 |
lemma CJNB_qe: |
1449 |
assumes qe: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))" |
|
1450 |
shows "\<forall>bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))" |
|
55768 | 1451 |
proof clarify |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1452 |
fix bs p |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1453 |
assume qfp: "qfree p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1454 |
let ?cjs = "conjuncts p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1455 |
let ?yes = "fst (partition bound0 ?cjs)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1456 |
let ?no = "snd (partition bound0 ?cjs)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1457 |
let ?cno = "list_conj ?no" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1458 |
let ?cyes = "list_conj ?yes" |
55768 | 1459 |
have part: "partition bound0 ?cjs = (?yes,?no)" |
1460 |
by simp |
|
1461 |
from partition_P[OF part] have "\<forall>q\<in> set ?yes. bound0 q" |
|
1462 |
by blast |
|
1463 |
then have yes_nb: "bound0 ?cyes" |
|
1464 |
by (simp add: list_conj_nb') |
|
1465 |
then have yes_qf: "qfree (decr0 ?cyes)" |
|
1466 |
by (simp add: decr0_qf) |
|
55754 | 1467 |
from conjuncts_qf[OF qfp] partition_set[OF part] |
55768 | 1468 |
have " \<forall>q\<in> set ?no. qfree q" |
1469 |
by auto |
|
1470 |
then have no_qf: "qfree ?cno" |
|
1471 |
by (simp add: list_conj_qf) |
|
1472 |
with qe have cno_qf:"qfree (qe ?cno)" |
|
1473 |
and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" |
|
1474 |
by blast+ |
|
55754 | 1475 |
from cno_qf yes_qf have qf: "qfree (CJNB qe p)" |
55768 | 1476 |
by (simp add: CJNB_def Let_def split_def) |
1477 |
{ |
|
1478 |
fix bs |
|
1479 |
from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" |
|
1480 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1481 |
also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1482 |
using partition_set[OF part] by auto |
55768 | 1483 |
finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" |
1484 |
using list_conj[of vs bs] by simp |
|
1485 |
} |
|
1486 |
then have "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" |
|
1487 |
by simp |
|
1488 |
also fix y have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1489 |
using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1490 |
also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))" |
33639
603320b93668
New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents:
33268
diff
changeset
|
1491 |
by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1492 |
also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1493 |
using qe[rule_format, OF no_qf] by auto |
55754 | 1494 |
finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1495 |
by (simp add: Let_def CJNB_def split_def) |
55768 | 1496 |
with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" |
1497 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1498 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1499 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1500 |
consts simpfm :: "fm \<Rightarrow> fm" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1501 |
recdef simpfm "measure fmsize" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1502 |
"simpfm (Lt t) = simplt (simptm t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1503 |
"simpfm (Le t) = simple (simptm t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1504 |
"simpfm (Eq t) = simpeq(simptm t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1505 |
"simpfm (NEq t) = simpneq(simptm t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1506 |
"simpfm (And p q) = conj (simpfm p) (simpfm q)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1507 |
"simpfm (Or p q) = disj (simpfm p) (simpfm q)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1508 |
"simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)" |
55768 | 1509 |
"simpfm (Iff p q) = |
1510 |
disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1511 |
"simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1512 |
"simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1513 |
"simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))" |
55768 | 1514 |
"simpfm (NOT (Iff p q)) = |
1515 |
disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1516 |
"simpfm (NOT (Eq t)) = simpneq t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1517 |
"simpfm (NOT (NEq t)) = simpeq t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1518 |
"simpfm (NOT (Le t)) = simplt (Neg t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1519 |
"simpfm (NOT (Lt t)) = simple (Neg t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1520 |
"simpfm (NOT (NOT p)) = simpfm p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1521 |
"simpfm (NOT T) = F" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1522 |
"simpfm (NOT F) = T" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1523 |
"simpfm p = p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1524 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1525 |
lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p" |
55768 | 1526 |
by (induct p arbitrary: bs rule: simpfm.induct) auto |
1527 |
||
1528 |
lemma simpfm_bound0: |
|
1529 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1530 |
shows "bound0 p \<Longrightarrow> bound0 (simpfm p)" |
55768 | 1531 |
by (induct p rule: simpfm.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1532 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1533 |
lemma lt_qf[simp]: "qfree (lt t)" |
55768 | 1534 |
apply (cases t) |
1535 |
apply (auto simp add: lt_def) |
|
1536 |
apply (case_tac poly) |
|
1537 |
apply auto |
|
1538 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1539 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1540 |
lemma le_qf[simp]: "qfree (le t)" |
55768 | 1541 |
apply (cases t) |
1542 |
apply (auto simp add: le_def) |
|
1543 |
apply (case_tac poly) |
|
1544 |
apply auto |
|
1545 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1546 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1547 |
lemma eq_qf[simp]: "qfree (eq t)" |
55768 | 1548 |
apply (cases t) |
1549 |
apply (auto simp add: eq_def) |
|
1550 |
apply (case_tac poly) |
|
1551 |
apply auto |
|
1552 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1553 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1554 |
lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1555 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1556 |
lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1557 |
lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1558 |
lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1559 |
lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1560 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1561 |
lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)" |
55768 | 1562 |
by (induct p rule: simpfm.induct) auto |
1563 |
||
1564 |
lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" |
|
1565 |
by (simp add: disj_def) |
|
1566 |
lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" |
|
1567 |
by (simp add: conj_def) |
|
1568 |
||
1569 |
lemma |
|
1570 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
55754 | 1571 |
shows "qfree p \<Longrightarrow> islin (simpfm p)" |
55768 | 1572 |
by (induct p rule: simpfm.induct) (simp_all add: conj_lin disj_lin) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1573 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1574 |
consts prep :: "fm \<Rightarrow> fm" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1575 |
recdef prep "measure fmsize" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1576 |
"prep (E T) = T" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1577 |
"prep (E F) = F" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1578 |
"prep (E (Or p q)) = disj (prep (E p)) (prep (E q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1579 |
"prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))" |
55754 | 1580 |
"prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1581 |
"prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1582 |
"prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1583 |
"prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1584 |
"prep (E p) = E (prep p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1585 |
"prep (A (And p q)) = conj (prep (A p)) (prep (A q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1586 |
"prep (A p) = prep (NOT (E (NOT p)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1587 |
"prep (NOT (NOT p)) = prep p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1588 |
"prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1589 |
"prep (NOT (A p)) = prep (E (NOT p))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1590 |
"prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1591 |
"prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1592 |
"prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1593 |
"prep (NOT p) = not (prep p)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1594 |
"prep (Or p q) = disj (prep p) (prep q)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1595 |
"prep (And p q) = conj (prep p) (prep q)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1596 |
"prep (Imp p q) = prep (Or (NOT p) q)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1597 |
"prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1598 |
"prep p = p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1599 |
(hints simp add: fmsize_pos) |
55768 | 1600 |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1601 |
lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p" |
55768 | 1602 |
by (induct p arbitrary: bs rule: prep.induct) auto |
1603 |
||
1604 |
||
1605 |
(* Generic quantifier elimination *) |
|
1606 |
function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" |
|
1607 |
where |
|
1608 |
"qelim (E p) = (\<lambda>qe. DJ (CJNB qe) (qelim p qe))" |
|
1609 |
| "qelim (A p) = (\<lambda>qe. not (qe ((qelim (NOT p) qe))))" |
|
1610 |
| "qelim (NOT p) = (\<lambda>qe. not (qelim p qe))" |
|
1611 |
| "qelim (And p q) = (\<lambda>qe. conj (qelim p qe) (qelim q qe))" |
|
1612 |
| "qelim (Or p q) = (\<lambda>qe. disj (qelim p qe) (qelim q qe))" |
|
1613 |
| "qelim (Imp p q) = (\<lambda>qe. imp (qelim p qe) (qelim q qe))" |
|
1614 |
| "qelim (Iff p q) = (\<lambda>qe. iff (qelim p qe) (qelim q qe))" |
|
1615 |
| "qelim p = (\<lambda>y. simpfm p)" |
|
41822 | 1616 |
by pat_completeness simp_all |
1617 |
termination by (relation "measure fmsize") auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1618 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1619 |
lemma qelim: |
55754 | 1620 |
assumes qe_inv: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1621 |
shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)" |
55768 | 1622 |
using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] |
1623 |
by (induct p rule: qelim.induct) auto |
|
1624 |
||
1625 |
||
1626 |
subsection {* Core Procedure *} |
|
1627 |
||
1628 |
fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) |
|
1629 |
where |
|
55754 | 1630 |
"minusinf (And p q) = conj (minusinf p) (minusinf q)" |
1631 |
| "minusinf (Or p q) = disj (minusinf p) (minusinf q)" |
|
41822 | 1632 |
| "minusinf (Eq (CNP 0 c e)) = conj (eq (CP c)) (eq e)" |
1633 |
| "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))" |
|
1634 |
| "minusinf (Lt (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))" |
|
1635 |
| "minusinf (Le (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))" |
|
1636 |
| "minusinf p = p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1637 |
|
55768 | 1638 |
fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) |
1639 |
where |
|
55754 | 1640 |
"plusinf (And p q) = conj (plusinf p) (plusinf q)" |
1641 |
| "plusinf (Or p q) = disj (plusinf p) (plusinf q)" |
|
41822 | 1642 |
| "plusinf (Eq (CNP 0 c e)) = conj (eq (CP c)) (eq e)" |
1643 |
| "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))" |
|
1644 |
| "plusinf (Lt (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))" |
|
1645 |
| "plusinf (Le (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))" |
|
1646 |
| "plusinf p = p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1647 |
|
55768 | 1648 |
lemma minusinf_inf: |
1649 |
assumes lp: "islin p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1650 |
shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1651 |
using lp |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1652 |
proof (induct p rule: minusinf.induct) |
55768 | 1653 |
case 1 |
1654 |
then show ?case |
|
1655 |
apply auto |
|
1656 |
apply (rule_tac x="min z za" in exI) |
|
1657 |
apply auto |
|
1658 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1659 |
next |
55768 | 1660 |
case 2 |
1661 |
then show ?case |
|
1662 |
apply auto |
|
1663 |
apply (rule_tac x="min z za" in exI) |
|
1664 |
apply auto |
|
1665 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1666 |
next |
55768 | 1667 |
case (3 c e) |
1668 |
then have nbe: "tmbound0 e" |
|
1669 |
by simp |
|
1670 |
from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1671 |
by simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1672 |
note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1673 |
let ?c = "Ipoly vs c" |
55768 | 1674 |
fix y |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1675 |
let ?e = "Itm vs (y#bs) e" |
55768 | 1676 |
have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith |
1677 |
moreover { |
|
1678 |
assume "?c = 0" |
|
1679 |
then have ?case |
|
1680 |
using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto |
|
1681 |
} |
|
1682 |
moreover { |
|
1683 |
assume cp: "?c > 0" |
|
1684 |
{ |
|
1685 |
fix x |
|
1686 |
assume xz: "x < -?e / ?c" |
|
1687 |
then have "?c * x < - ?e" |
|
1688 |
using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1689 |
by (simp add: mult.commute) |
55768 | 1690 |
then have "?c * x + ?e < 0" |
1691 |
by simp |
|
1692 |
then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))" |
|
1693 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto |
|
1694 |
} |
|
1695 |
then have ?case by auto |
|
1696 |
} |
|
1697 |
moreover { |
|
1698 |
assume cp: "?c < 0" |
|
1699 |
{ |
|
1700 |
fix x |
|
1701 |
assume xz: "x < -?e / ?c" |
|
1702 |
then have "?c * x > - ?e" |
|
1703 |
using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1704 |
by (simp add: mult.commute) |
55768 | 1705 |
then have "?c * x + ?e > 0" |
1706 |
by simp |
|
1707 |
then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))" |
|
1708 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto |
|
1709 |
} |
|
1710 |
then have ?case by auto |
|
1711 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1712 |
ultimately show ?case by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1713 |
next |
55768 | 1714 |
case (4 c e) |
1715 |
then have nbe: "tmbound0 e" |
|
1716 |
by simp |
|
1717 |
from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1718 |
by simp_all |
|
1719 |
note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1720 |
let ?c = "Ipoly vs c" |
55768 | 1721 |
fix y |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1722 |
let ?e = "Itm vs (y#bs) e" |
55768 | 1723 |
have "?c=0 \<or> ?c > 0 \<or> ?c < 0" |
1724 |
by arith |
|
1725 |
moreover { |
|
1726 |
assume "?c = 0" |
|
1727 |
then have ?case |
|
1728 |
using eqs by auto |
|
1729 |
} |
|
1730 |
moreover { |
|
1731 |
assume cp: "?c > 0" |
|
1732 |
{ |
|
1733 |
fix x |
|
1734 |
assume xz: "x < -?e / ?c" |
|
1735 |
then have "?c * x < - ?e" |
|
1736 |
using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1737 |
by (simp add: mult.commute) |
55768 | 1738 |
then have "?c * x + ?e < 0" |
1739 |
by simp |
|
1740 |
then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))" |
|
1741 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto |
|
1742 |
} |
|
1743 |
then have ?case by auto |
|
1744 |
} |
|
1745 |
moreover { |
|
1746 |
assume cp: "?c < 0" |
|
1747 |
{ |
|
1748 |
fix x |
|
1749 |
assume xz: "x < -?e / ?c" |
|
1750 |
then have "?c * x > - ?e" |
|
1751 |
using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1752 |
by (simp add: mult.commute) |
55768 | 1753 |
then have "?c * x + ?e > 0" |
1754 |
by simp |
|
1755 |
then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))" |
|
1756 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto |
|
1757 |
} |
|
1758 |
then have ?case by auto |
|
1759 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1760 |
ultimately show ?case by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1761 |
next |
55768 | 1762 |
case (5 c e) |
1763 |
then have nbe: "tmbound0 e" |
|
1764 |
by simp |
|
1765 |
from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1766 |
by simp_all |
|
1767 |
then have nc': "allpolys isnpoly (CP (~\<^sub>p c))" |
|
1768 |
by (simp add: polyneg_norm) |
|
1769 |
note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a] |
|
1770 |
let ?c = "Ipoly vs c" |
|
1771 |
fix y |
|
1772 |
let ?e = "Itm vs (y#bs) e" |
|
1773 |
have "?c=0 \<or> ?c > 0 \<or> ?c < 0" |
|
1774 |
by arith |
|
1775 |
moreover { |
|
1776 |
assume "?c = 0" |
|
1777 |
then have ?case using eqs by auto |
|
1778 |
} |
|
1779 |
moreover { |
|
1780 |
assume cp: "?c > 0" |
|
1781 |
{ |
|
1782 |
fix x |
|
1783 |
assume xz: "x < -?e / ?c" |
|
1784 |
then have "?c * x < - ?e" |
|
1785 |
using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1786 |
by (simp add: mult.commute) |
55768 | 1787 |
then have "?c * x + ?e < 0" by simp |
1788 |
then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))" |
|
1789 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto |
|
1790 |
} |
|
1791 |
then have ?case by auto |
|
1792 |
} |
|
1793 |
moreover { |
|
1794 |
assume cp: "?c < 0" |
|
1795 |
{ |
|
1796 |
fix x |
|
1797 |
assume xz: "x < -?e / ?c" |
|
1798 |
then have "?c * x > - ?e" |
|
1799 |
using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1800 |
by (simp add: mult.commute) |
55768 | 1801 |
then have "?c * x + ?e > 0" |
1802 |
by simp |
|
1803 |
then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))" |
|
1804 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto |
|
1805 |
} |
|
1806 |
then have ?case by auto |
|
1807 |
} |
|
1808 |
ultimately show ?case by blast |
|
1809 |
next |
|
1810 |
case (6 c e) |
|
1811 |
then have nbe: "tmbound0 e" |
|
1812 |
by simp |
|
1813 |
from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1814 |
by simp_all |
|
1815 |
then have nc': "allpolys isnpoly (CP (~\<^sub>p c))" |
|
1816 |
by (simp add: polyneg_norm) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1817 |
note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a] |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1818 |
let ?c = "Ipoly vs c" |
55768 | 1819 |
fix y |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1820 |
let ?e = "Itm vs (y#bs) e" |
55768 | 1821 |
have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith |
1822 |
moreover { |
|
1823 |
assume "?c = 0" |
|
1824 |
then have ?case using eqs by auto |
|
1825 |
} |
|
1826 |
moreover { |
|
1827 |
assume cp: "?c > 0" |
|
1828 |
{ |
|
1829 |
fix x |
|
1830 |
assume xz: "x < -?e / ?c" |
|
1831 |
then have "?c * x < - ?e" |
|
1832 |
using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1833 |
by (simp add: mult.commute) |
55768 | 1834 |
then have "?c * x + ?e < 0" |
1835 |
by simp |
|
1836 |
then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))" |
|
1837 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs |
|
1838 |
by auto |
|
1839 |
} |
|
1840 |
then have ?case by auto |
|
1841 |
} |
|
1842 |
moreover { |
|
1843 |
assume cp: "?c < 0" |
|
1844 |
{ |
|
1845 |
fix x |
|
1846 |
assume xz: "x < -?e / ?c" |
|
1847 |
then have "?c * x > - ?e" |
|
1848 |
using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1849 |
by (simp add: mult.commute) |
55768 | 1850 |
then have "?c * x + ?e > 0" |
1851 |
by simp |
|
1852 |
then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))" |
|
1853 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs |
|
1854 |
by auto |
|
1855 |
} |
|
1856 |
then have ?case by auto |
|
1857 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1858 |
ultimately show ?case by blast |
55768 | 1859 |
qed auto |
1860 |
||
1861 |
lemma plusinf_inf: |
|
1862 |
assumes lp: "islin p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1863 |
shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1864 |
using lp |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1865 |
proof (induct p rule: plusinf.induct) |
55768 | 1866 |
case 1 |
1867 |
then show ?case |
|
1868 |
apply auto |
|
1869 |
apply (rule_tac x="max z za" in exI) |
|
1870 |
apply auto |
|
1871 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1872 |
next |
55768 | 1873 |
case 2 |
1874 |
then show ?case |
|
1875 |
apply auto |
|
1876 |
apply (rule_tac x="max z za" in exI) |
|
1877 |
apply auto |
|
1878 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1879 |
next |
55768 | 1880 |
case (3 c e) |
1881 |
then have nbe: "tmbound0 e" |
|
1882 |
by simp |
|
1883 |
from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1884 |
by simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1885 |
note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1886 |
let ?c = "Ipoly vs c" |
55768 | 1887 |
fix y |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1888 |
let ?e = "Itm vs (y#bs) e" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1889 |
have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith |
55768 | 1890 |
moreover { |
1891 |
assume "?c = 0" |
|
1892 |
then have ?case |
|
1893 |
using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto |
|
1894 |
} |
|
1895 |
moreover { |
|
1896 |
assume cp: "?c > 0" |
|
1897 |
{ |
|
1898 |
fix x |
|
1899 |
assume xz: "x > -?e / ?c" |
|
1900 |
then have "?c * x > - ?e" |
|
1901 |
using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1902 |
by (simp add: mult.commute) |
55768 | 1903 |
then have "?c * x + ?e > 0" |
1904 |
by simp |
|
1905 |
then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))" |
|
1906 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto |
|
1907 |
} |
|
1908 |
then have ?case by auto |
|
1909 |
} |
|
1910 |
moreover { |
|
1911 |
assume cp: "?c < 0" |
|
1912 |
{ |
|
1913 |
fix x |
|
1914 |
assume xz: "x > -?e / ?c" |
|
1915 |
then have "?c * x < - ?e" |
|
1916 |
using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1917 |
by (simp add: mult.commute) |
55768 | 1918 |
then have "?c * x + ?e < 0" by simp |
1919 |
then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))" |
|
1920 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto |
|
1921 |
} |
|
1922 |
then have ?case by auto |
|
1923 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1924 |
ultimately show ?case by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1925 |
next |
55768 | 1926 |
case (4 c e) |
1927 |
then have nbe: "tmbound0 e" |
|
1928 |
by simp |
|
1929 |
from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1930 |
by simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1931 |
note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1932 |
let ?c = "Ipoly vs c" |
55768 | 1933 |
fix y |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1934 |
let ?e = "Itm vs (y#bs) e" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1935 |
have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith |
55768 | 1936 |
moreover { |
1937 |
assume "?c = 0" |
|
1938 |
then have ?case using eqs by auto |
|
1939 |
} |
|
1940 |
moreover { |
|
1941 |
assume cp: "?c > 0" |
|
1942 |
{ |
|
1943 |
fix x |
|
1944 |
assume xz: "x > -?e / ?c" |
|
1945 |
then have "?c * x > - ?e" |
|
1946 |
using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1947 |
by (simp add: mult.commute) |
55768 | 1948 |
then have "?c * x + ?e > 0" |
1949 |
by simp |
|
1950 |
then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))" |
|
1951 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto |
|
1952 |
} |
|
1953 |
then have ?case by auto |
|
1954 |
} |
|
1955 |
moreover { |
|
1956 |
assume cp: "?c < 0" |
|
1957 |
{ |
|
1958 |
fix x |
|
1959 |
assume xz: "x > -?e / ?c" |
|
1960 |
then have "?c * x < - ?e" |
|
1961 |
using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1962 |
by (simp add: mult.commute) |
55768 | 1963 |
then have "?c * x + ?e < 0" |
1964 |
by simp |
|
1965 |
then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))" |
|
1966 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto |
|
1967 |
} |
|
1968 |
then have ?case by auto |
|
1969 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1970 |
ultimately show ?case by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
1971 |
next |
55768 | 1972 |
case (5 c e) |
1973 |
then have nbe: "tmbound0 e" |
|
1974 |
by simp |
|
1975 |
from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
1976 |
by simp_all |
|
1977 |
then have nc': "allpolys isnpoly (CP (~\<^sub>p c))" |
|
1978 |
by (simp add: polyneg_norm) |
|
1979 |
note eqs = lt[OF nc(1), where ?'a = 'a] lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a] |
|
1980 |
let ?c = "Ipoly vs c" |
|
1981 |
fix y |
|
1982 |
let ?e = "Itm vs (y#bs) e" |
|
1983 |
have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith |
|
1984 |
moreover { |
|
1985 |
assume "?c = 0" |
|
1986 |
then have ?case using eqs by auto |
|
1987 |
} |
|
1988 |
moreover { |
|
1989 |
assume cp: "?c > 0" |
|
1990 |
{ |
|
1991 |
fix x |
|
1992 |
assume xz: "x > -?e / ?c" |
|
1993 |
then have "?c * x > - ?e" |
|
1994 |
using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
1995 |
by (simp add: mult.commute) |
55768 | 1996 |
then have "?c * x + ?e > 0" |
1997 |
by simp |
|
1998 |
then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))" |
|
1999 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto |
|
2000 |
} |
|
2001 |
then have ?case by auto |
|
2002 |
} |
|
2003 |
moreover { |
|
2004 |
assume cp: "?c < 0" |
|
2005 |
{ |
|
2006 |
fix x |
|
2007 |
assume xz: "x > -?e / ?c" |
|
2008 |
then have "?c * x < - ?e" |
|
2009 |
using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
2010 |
by (simp add: mult.commute) |
55768 | 2011 |
then have "?c * x + ?e < 0" |
2012 |
by simp |
|
2013 |
then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))" |
|
2014 |
using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto |
|
2015 |
} |
|
2016 |
then have ?case by auto |
|
2017 |
} |
|
2018 |
ultimately show ?case by blast |
|
2019 |
next |
|
2020 |
case (6 c e) |
|
2021 |
then have nbe: "tmbound0 e" |
|
2022 |
by simp |
|
2023 |
from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" |
|
2024 |
by simp_all |
|
2025 |
then have nc': "allpolys isnpoly (CP (~\<^sub>p c))" |
|
2026 |
by (simp add: polyneg_norm) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2027 |
note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a] |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2028 |
let ?c = "Ipoly vs c" |
55768 | 2029 |
fix y |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2030 |
let ?e = "Itm vs (y#bs) e" |
55768 | 2031 |
have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith |
2032 |
moreover { |
|
2033 |
assume "?c = 0" |
|
2034 |
then have ?case using eqs by auto |
|
2035 |
} |
|
2036 |
moreover { |
|
2037 |
assume cp: "?c > 0" |
|
2038 |
{ |
|
2039 |
fix x |
|
2040 |
assume xz: "x > -?e / ?c" |
|
2041 |
then have "?c * x > - ?e" |
|
2042 |
using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
2043 |
by (simp add: mult.commute) |
55768 | 2044 |
then have "?c * x + ?e > 0" |
2045 |
by simp |
|
2046 |
then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))" |
|
2047 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto |
|
2048 |
} |
|
2049 |
then have ?case by auto |
|
2050 |
} |
|
2051 |
moreover { |
|
2052 |
assume cp: "?c < 0" |
|
2053 |
{ |
|
2054 |
fix x |
|
2055 |
assume xz: "x > -?e / ?c" |
|
2056 |
then have "?c * x < - ?e" |
|
2057 |
using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
2058 |
by (simp add: mult.commute) |
55768 | 2059 |
then have "?c * x + ?e < 0" |
2060 |
by simp |
|
2061 |
then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))" |
|
2062 |
using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto |
|
2063 |
} |
|
2064 |
then have ?case by auto |
|
2065 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2066 |
ultimately show ?case by blast |
55768 | 2067 |
qed auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2068 |
|
55754 | 2069 |
lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)" |
55768 | 2070 |
by (induct p rule: minusinf.induct) (auto simp add: eq_nb lt_nb le_nb) |
2071 |
||
55754 | 2072 |
lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)" |
55768 | 2073 |
by (induct p rule: minusinf.induct) (auto simp add: eq_nb lt_nb le_nb) |
2074 |
||
2075 |
lemma minusinf_ex: |
|
2076 |
assumes lp: "islin p" |
|
2077 |
and ex: "Ifm vs (x#bs) (minusinf p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2078 |
shows "\<exists>x. Ifm vs (x#bs) p" |
55768 | 2079 |
proof - |
2080 |
from bound0_I [OF minusinf_nb[OF lp], where bs ="bs"] ex |
|
2081 |
have th: "\<forall>x. Ifm vs (x#bs) (minusinf p)" |
|
2082 |
by auto |
|
55754 | 2083 |
from minusinf_inf[OF lp, where bs="bs"] |
55768 | 2084 |
obtain z where z: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" |
2085 |
by blast |
|
2086 |
from th have "Ifm vs ((z - 1)#bs) (minusinf p)" |
|
2087 |
by simp |
|
2088 |
moreover have "z - 1 < z" |
|
2089 |
by simp |
|
2090 |
ultimately show ?thesis |
|
2091 |
using z by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2092 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2093 |
|
55768 | 2094 |
lemma plusinf_ex: |
2095 |
assumes lp: "islin p" |
|
2096 |
and ex: "Ifm vs (x#bs) (plusinf p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2097 |
shows "\<exists>x. Ifm vs (x#bs) p" |
55768 | 2098 |
proof - |
2099 |
from bound0_I [OF plusinf_nb[OF lp], where bs ="bs"] ex |
|
2100 |
have th: "\<forall>x. Ifm vs (x#bs) (plusinf p)" |
|
2101 |
by auto |
|
55754 | 2102 |
from plusinf_inf[OF lp, where bs="bs"] |
55768 | 2103 |
obtain z where z: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" |
2104 |
by blast |
|
2105 |
from th have "Ifm vs ((z + 1)#bs) (plusinf p)" |
|
2106 |
by simp |
|
2107 |
moreover have "z + 1 > z" |
|
2108 |
by simp |
|
2109 |
ultimately show ?thesis |
|
2110 |
using z by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2111 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2112 |
|
55768 | 2113 |
fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" |
2114 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2115 |
"uset (And p q) = uset p @ uset q" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2116 |
| "uset (Or p q) = uset p @ uset q" |
55768 | 2117 |
| "uset (Eq (CNP 0 a e)) = [(a, e)]" |
2118 |
| "uset (Le (CNP 0 a e)) = [(a, e)]" |
|
2119 |
| "uset (Lt (CNP 0 a e)) = [(a, e)]" |
|
2120 |
| "uset (NEq (CNP 0 a e)) = [(a, e)]" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2121 |
| "uset p = []" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2122 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2123 |
lemma uset_l: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2124 |
assumes lp: "islin p" |
55754 | 2125 |
shows "\<forall>(c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s" |
55768 | 2126 |
using lp by (induct p rule: uset.induct) auto |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2127 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2128 |
lemma minusinf_uset0: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2129 |
assumes lp: "islin p" |
55768 | 2130 |
and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" |
2131 |
and ex: "Ifm vs (x#bs) p" (is "?I x p") |
|
2132 |
shows "\<exists>(c, s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c" |
|
2133 |
proof - |
|
2134 |
have "\<exists>(c, s) \<in> set (uset p). |
|
2135 |
Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s \<or> |
|
2136 |
Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2137 |
using lp nmi ex |
55768 | 2138 |
apply (induct p rule: minusinf.induct) |
2139 |
apply (auto simp add: eq le lt polyneg_norm) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2140 |
apply (auto simp add: linorder_not_less order_le_less) |
55754 | 2141 |
done |
55768 | 2142 |
then obtain c s where csU: "(c, s) \<in> set (uset p)" |
2143 |
and x: "(Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s) \<or> |
|
2144 |
(Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)" by blast |
|
2145 |
then have "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2146 |
using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x] |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
2147 |
by (auto simp add: mult.commute) |
55768 | 2148 |
then show ?thesis |
2149 |
using csU by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2150 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2151 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2152 |
lemma minusinf_uset: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2153 |
assumes lp: "islin p" |
55768 | 2154 |
and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))" |
2155 |
and ex: "Ifm vs (x#bs) p" (is "?I x p") |
|
55754 | 2156 |
shows "\<exists>(c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c" |
55768 | 2157 |
proof - |
2158 |
from nmi have nmi': "\<not> Ifm vs (x#bs) (minusinf p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2159 |
by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a]) |
55754 | 2160 |
from minusinf_uset0[OF lp nmi' ex] |
55768 | 2161 |
obtain c s where csU: "(c,s) \<in> set (uset p)" |
2162 |
and th: "x \<ge> - Itm vs (x#bs) s / Ipoly vs c" |
|
2163 |
by blast |
|
2164 |
from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" |
|
2165 |
by simp |
|
2166 |
from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis |
|
2167 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2168 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2169 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2170 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2171 |
lemma plusinf_uset0: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2172 |
assumes lp: "islin p" |
55768 | 2173 |
and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))" |
2174 |
and ex: "Ifm vs (x#bs) p" (is "?I x p") |
|
2175 |
shows "\<exists>(c, s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2176 |
proof- |
55768 | 2177 |
have "\<exists>(c, s) \<in> set (uset p). |
2178 |
Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s \<or> |
|
2179 |
Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2180 |
using lp nmi ex |
55768 | 2181 |
apply (induct p rule: minusinf.induct) |
2182 |
apply (auto simp add: eq le lt polyneg_norm) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2183 |
apply (auto simp add: linorder_not_less order_le_less) |
55754 | 2184 |
done |
55768 | 2185 |
then obtain c s where csU: "(c, s) \<in> set (uset p)" |
2186 |
and x: "Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s \<or> |
|
2187 |
Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s" |
|
2188 |
by blast |
|
2189 |
then have "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2190 |
using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"] |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
2191 |
by (auto simp add: mult.commute) |
55768 | 2192 |
then show ?thesis |
2193 |
using csU by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2194 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2195 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2196 |
lemma plusinf_uset: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2197 |
assumes lp: "islin p" |
55768 | 2198 |
and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))" |
2199 |
and ex: "Ifm vs (x#bs) p" (is "?I x p") |
|
55754 | 2200 |
shows "\<exists>(c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c" |
55768 | 2201 |
proof - |
55754 | 2202 |
from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2203 |
by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a]) |
55754 | 2204 |
from plusinf_uset0[OF lp nmi' ex] |
55768 | 2205 |
obtain c s where csU: "(c,s) \<in> set (uset p)" |
2206 |
and th: "x \<le> - Itm vs (x#bs) s / Ipoly vs c" |
|
2207 |
by blast |
|
2208 |
from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" |
|
2209 |
by simp |
|
2210 |
from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis |
|
2211 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2212 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2213 |
|
55754 | 2214 |
lemma lin_dense: |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2215 |
assumes lp: "islin p" |
55768 | 2216 |
and noS: "\<forall>t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda>(c,t). - Itm vs (x#bs) t / Ipoly vs c) ` set (uset p)" |
2217 |
(is "\<forall>t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda>(c,t). - ?Nt x t / ?N c) ` ?U p") |
|
2218 |
and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p" |
|
2219 |
and ly: "l < y" and yu: "y < u" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2220 |
shows "Ifm vs (y#bs) p" |
55768 | 2221 |
using lp px noS |
55754 | 2222 |
proof (induct p rule: islin.induct) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2223 |
case (5 c s) |
55754 | 2224 |
from "5.prems" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2225 |
have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2226 |
and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))" |
55768 | 2227 |
and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
2228 |
by simp_all |
|
2229 |
from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
|
2230 |
by simp |
|
2231 |
then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" |
|
2232 |
by auto |
|
2233 |
have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" |
|
2234 |
by dlo |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2235 |
moreover |
55768 | 2236 |
{ |
2237 |
assume "?N c = 0" |
|
2238 |
then have ?case |
|
2239 |
using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2240 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2241 |
moreover |
55768 | 2242 |
{ |
2243 |
assume c: "?N c > 0" |
|
2244 |
from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"] |
|
2245 |
have px': "x < - ?Nt x s / ?N c" |
|
2246 |
by (auto simp add: not_less field_simps) |
|
2247 |
{ |
|
2248 |
assume y: "y < - ?Nt x s / ?N c" |
|
2249 |
then have "y * ?N c < - ?Nt x s" |
|
2250 |
by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric]) |
|
2251 |
then have "?N c * y + ?Nt x s < 0" |
|
2252 |
by (simp add: field_simps) |
|
2253 |
then have ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] |
|
2254 |
by simp |
|
2255 |
} |
|
2256 |
moreover |
|
2257 |
{ |
|
2258 |
assume y: "y > -?Nt x s / ?N c" |
|
2259 |
with yu have eu: "u > - ?Nt x s / ?N c" |
|
2260 |
by auto |
|
2261 |
with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" |
|
2262 |
by (cases "- ?Nt x s / ?N c > l") auto |
|
2263 |
with lx px' have False |
|
2264 |
by simp |
|
2265 |
then have ?case .. |
|
2266 |
} |
|
2267 |
ultimately have ?case |
|
2268 |
using ycs by blast |
|
2269 |
} |
|
2270 |
moreover |
|
2271 |
{ |
|
2272 |
assume c: "?N c < 0" |
|
2273 |
from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"] |
|
2274 |
have px': "x > - ?Nt x s / ?N c" |
|
2275 |
by (auto simp add: not_less field_simps) |
|
2276 |
{ |
|
2277 |
assume y: "y > - ?Nt x s / ?N c" |
|
2278 |
then have "y * ?N c < - ?Nt x s" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
2279 |
by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric]) |
55768 | 2280 |
then have "?N c * y + ?Nt x s < 0" |
2281 |
by (simp add: field_simps) |
|
2282 |
then have ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] |
|
2283 |
by simp |
|
2284 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2285 |
moreover |
55768 | 2286 |
{ |
2287 |
assume y: "y < -?Nt x s / ?N c" |
|
2288 |
with ly have eu: "l < - ?Nt x s / ?N c" |
|
2289 |
by auto |
|
2290 |
with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" |
|
2291 |
by (cases "- ?Nt x s / ?N c < u") auto |
|
2292 |
with xu px' have False |
|
2293 |
by simp |
|
2294 |
then have ?case .. |
|
2295 |
} |
|
2296 |
ultimately have ?case |
|
2297 |
using ycs by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2298 |
} |
55768 | 2299 |
ultimately show ?case |
2300 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2301 |
next |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2302 |
case (6 c s) |
55754 | 2303 |
from "6.prems" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2304 |
have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2305 |
and px: "Ifm vs (x # bs) (Le (CNP 0 c s))" |
55768 | 2306 |
and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
2307 |
by simp_all |
|
2308 |
from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
|
2309 |
by simp |
|
2310 |
then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" |
|
2311 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2312 |
have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2313 |
moreover |
55768 | 2314 |
{ |
2315 |
assume "?N c = 0" |
|
2316 |
then have ?case |
|
2317 |
using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2318 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2319 |
moreover |
55768 | 2320 |
{ |
2321 |
assume c: "?N c > 0" |
|
2322 |
from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"] |
|
2323 |
have px': "x \<le> - ?Nt x s / ?N c" |
|
2324 |
by (simp add: not_less field_simps) |
|
2325 |
{ |
|
2326 |
assume y: "y < - ?Nt x s / ?N c" |
|
2327 |
then have "y * ?N c < - ?Nt x s" |
|
2328 |
by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric]) |
|
2329 |
then have "?N c * y + ?Nt x s < 0" |
|
2330 |
by (simp add: field_simps) |
|
2331 |
then have ?case |
|
2332 |
using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp |
|
2333 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2334 |
moreover |
55768 | 2335 |
{ |
2336 |
assume y: "y > -?Nt x s / ?N c" |
|
2337 |
with yu have eu: "u > - ?Nt x s / ?N c" |
|
2338 |
by auto |
|
2339 |
with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" |
|
2340 |
by (cases "- ?Nt x s / ?N c > l") auto |
|
2341 |
with lx px' have False |
|
2342 |
by simp |
|
2343 |
then have ?case .. |
|
2344 |
} |
|
2345 |
ultimately have ?case |
|
2346 |
using ycs by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2347 |
} |
55768 | 2348 |
moreover |
2349 |
{ |
|
2350 |
assume c: "?N c < 0" |
|
2351 |
from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"] |
|
2352 |
have px': "x >= - ?Nt x s / ?N c" |
|
2353 |
by (simp add: field_simps) |
|
2354 |
{ |
|
2355 |
assume y: "y > - ?Nt x s / ?N c" |
|
2356 |
then have "y * ?N c < - ?Nt x s" |
|
2357 |
by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric]) |
|
2358 |
then have "?N c * y + ?Nt x s < 0" |
|
2359 |
by (simp add: field_simps) |
|
2360 |
then have ?case |
|
2361 |
using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp |
|
2362 |
} |
|
2363 |
moreover |
|
2364 |
{ |
|
2365 |
assume y: "y < -?Nt x s / ?N c" |
|
2366 |
with ly have eu: "l < - ?Nt x s / ?N c" |
|
2367 |
by auto |
|
2368 |
with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" |
|
2369 |
by (cases "- ?Nt x s / ?N c < u") auto |
|
2370 |
with xu px' have False by simp |
|
2371 |
then have ?case .. |
|
2372 |
} |
|
2373 |
ultimately have ?case |
|
2374 |
using ycs by blast |
|
2375 |
} |
|
2376 |
ultimately show ?case |
|
2377 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2378 |
next |
55768 | 2379 |
case (3 c s) |
55754 | 2380 |
from "3.prems" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2381 |
have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2382 |
and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))" |
55768 | 2383 |
and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
2384 |
by simp_all |
|
2385 |
from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
|
2386 |
by simp |
|
2387 |
then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" |
|
2388 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2389 |
have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2390 |
moreover |
55768 | 2391 |
{ |
2392 |
assume "?N c = 0" |
|
2393 |
then have ?case |
|
2394 |
using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2395 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2396 |
moreover |
55768 | 2397 |
{ |
2398 |
assume c: "?N c > 0" |
|
2399 |
then have cnz: "?N c \<noteq> 0" |
|
2400 |
by simp |
|
2401 |
from px eq_divide_eq[of "x" "-?Nt x s" "?N c"] cnz |
|
2402 |
have px': "x = - ?Nt x s / ?N c" |
|
2403 |
by (simp add: field_simps) |
|
2404 |
{ |
|
2405 |
assume y: "y < -?Nt x s / ?N c" |
|
2406 |
with ly have eu: "l < - ?Nt x s / ?N c" |
|
2407 |
by auto |
|
2408 |
with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" |
|
2409 |
by (cases "- ?Nt x s / ?N c < u") auto |
|
2410 |
with xu px' have False by simp |
|
2411 |
then have ?case .. |
|
2412 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2413 |
moreover |
55768 | 2414 |
{ |
2415 |
assume y: "y > -?Nt x s / ?N c" |
|
2416 |
with yu have eu: "u > - ?Nt x s / ?N c" |
|
2417 |
by auto |
|
2418 |
with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" |
|
2419 |
by (cases "- ?Nt x s / ?N c > l") auto |
|
2420 |
with lx px' have False by simp |
|
2421 |
then have ?case .. |
|
2422 |
} |
|
2423 |
ultimately have ?case |
|
2424 |
using ycs by blast |
|
2425 |
} |
|
2426 |
moreover |
|
2427 |
{ |
|
2428 |
assume c: "?N c < 0" |
|
2429 |
then have cnz: "?N c \<noteq> 0" |
|
2430 |
by simp |
|
2431 |
from px eq_divide_eq[of "x" "-?Nt x s" "?N c"] cnz |
|
2432 |
have px': "x = - ?Nt x s / ?N c" |
|
2433 |
by (simp add: field_simps) |
|
2434 |
{ |
|
2435 |
assume y: "y < -?Nt x s / ?N c" |
|
2436 |
with ly have eu: "l < - ?Nt x s / ?N c" |
|
2437 |
by auto |
|
2438 |
with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" |
|
2439 |
by (cases "- ?Nt x s / ?N c < u") auto |
|
2440 |
with xu px' have False by simp |
|
2441 |
then have ?case .. |
|
2442 |
} |
|
2443 |
moreover |
|
2444 |
{ |
|
2445 |
assume y: "y > -?Nt x s / ?N c" |
|
2446 |
with yu have eu: "u > - ?Nt x s / ?N c" |
|
2447 |
by auto |
|
2448 |
with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" |
|
2449 |
by (cases "- ?Nt x s / ?N c > l") auto |
|
2450 |
with lx px' have False by simp |
|
2451 |
then have ?case .. |
|
2452 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2453 |
ultimately have ?case using ycs by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2454 |
} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2455 |
ultimately show ?case by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2456 |
next |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2457 |
case (4 c s) |
55754 | 2458 |
from "4.prems" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2459 |
have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2460 |
and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))" |
55768 | 2461 |
and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
2462 |
by simp_all |
|
2463 |
from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
|
2464 |
by simp |
|
2465 |
then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" |
|
2466 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2467 |
have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2468 |
moreover |
55768 | 2469 |
{ |
2470 |
assume "?N c = 0" |
|
2471 |
then have ?case |
|
2472 |
using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) |
|
2473 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2474 |
moreover |
55768 | 2475 |
{ |
2476 |
assume c: "?N c \<noteq> 0" |
|
2477 |
from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] |
|
2478 |
have ?case |
|
2479 |
by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) |
|
2480 |
} |
|
2481 |
ultimately show ?case |
|
2482 |
by blast |
|
41842 | 2483 |
qed (auto simp add: tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"]) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2484 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2485 |
lemma inf_uset: |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2486 |
assumes lp: "islin p" |
55768 | 2487 |
and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))") |
2488 |
and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))") |
|
2489 |
and ex: "\<exists>x. Ifm vs (x#bs) p" (is "\<exists>x. ?I x p") |
|
2490 |
shows "\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p). |
|
2491 |
?I ((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2) p" |
|
2492 |
proof - |
|
2493 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2494 |
let ?N = "Ipoly vs" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2495 |
let ?U = "set (uset p)" |
55768 | 2496 |
from ex obtain a where pa: "?I a p" |
2497 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2498 |
from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi |
55768 | 2499 |
have nmi': "\<not> (?I a (?M p))" |
2500 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2501 |
from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi |
55768 | 2502 |
have npi': "\<not> (?I a (?P p))" |
2503 |
by simp |
|
55754 | 2504 |
have "\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). ?I ((- ?Nt a t/?N c + - ?Nt a s /?N d) / 2) p" |
55768 | 2505 |
proof - |
2506 |
let ?M = "(\<lambda>(c,t). - ?Nt a t / ?N c) ` ?U" |
|
2507 |
have fM: "finite ?M" |
|
2508 |
by auto |
|
55754 | 2509 |
from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa] |
55768 | 2510 |
have "\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p). |
2511 |
a \<le> - ?Nt x t / ?N c \<and> a \<ge> - ?Nt x s / ?N d" |
|
2512 |
by blast |
|
2513 |
then obtain "c" "t" "d" "s" |
|
2514 |
where ctU: "(c,t) \<in> ?U" |
|
2515 |
and dsU: "(d,s) \<in> ?U" |
|
2516 |
and xs1: "a \<le> - ?Nt x s / ?N d" |
|
2517 |
and tx1: "a \<ge> - ?Nt x t / ?N c" |
|
2518 |
by blast |
|
55754 | 2519 |
from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 |
55768 | 2520 |
have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" |
2521 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2522 |
from ctU have Mne: "?M \<noteq> {}" by auto |
55768 | 2523 |
then have Une: "?U \<noteq> {}" by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2524 |
let ?l = "Min ?M" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2525 |
let ?u = "Max ?M" |
55768 | 2526 |
have linM: "?l \<in> ?M" |
2527 |
using fM Mne by simp |
|
2528 |
have uinM: "?u \<in> ?M" |
|
2529 |
using fM Mne by simp |
|
2530 |
have ctM: "- ?Nt a t / ?N c \<in> ?M" |
|
2531 |
using ctU by auto |
|
2532 |
have dsM: "- ?Nt a s / ?N d \<in> ?M" |
|
2533 |
using dsU by auto |
|
2534 |
have lM: "\<forall>t\<in> ?M. ?l \<le> t" |
|
2535 |
using Mne fM by auto |
|
2536 |
have Mu: "\<forall>t\<in> ?M. t \<le> ?u" |
|
2537 |
using Mne fM by auto |
|
2538 |
have "?l \<le> - ?Nt a t / ?N c" |
|
2539 |
using ctM Mne by simp |
|
2540 |
then have lx: "?l \<le> a" |
|
2541 |
using tx by simp |
|
2542 |
have "- ?Nt a s / ?N d \<le> ?u" |
|
2543 |
using dsM Mne by simp |
|
2544 |
then have xu: "a \<le> ?u" |
|
2545 |
using xs by simp |
|
2546 |
from finite_set_intervals2[where P="\<lambda>x. ?I x p",OF pa lx xu linM uinM fM lM Mu] |
|
55754 | 2547 |
have "(\<exists>s\<in> ?M. ?I s p) \<or> |
2548 |
(\<exists>t1\<in> ?M. \<exists>t2 \<in> ?M. (\<forall>y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" . |
|
55768 | 2549 |
moreover { |
2550 |
fix u |
|
2551 |
assume um: "u\<in> ?M" |
|
2552 |
and pu: "?I u p" |
|
2553 |
then have "\<exists>(nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" |
|
2554 |
by auto |
|
2555 |
then obtain tu nu where tuU: "(nu, tu) \<in> ?U" |
|
2556 |
and tuu:"u= - ?Nt a tu / ?N nu" |
|
2557 |
by blast |
|
2558 |
from pu tuu have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / 2) p" |
|
2559 |
by simp |
|
2560 |
with tuU have ?thesis by blast |
|
2561 |
} |
|
2562 |
moreover { |
|
55754 | 2563 |
assume "\<exists>t1\<in> ?M. \<exists>t2 \<in> ?M. (\<forall>y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p" |
55768 | 2564 |
then obtain t1 t2 |
2565 |
where t1M: "t1 \<in> ?M" |
|
2566 |
and t2M: "t2\<in> ?M" |
|
2567 |
and noM: "\<forall>y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" |
|
2568 |
and t1x: "t1 < a" |
|
2569 |
and xt2: "a < t2" |
|
2570 |
and px: "?I a p" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
2571 |
by blast |
55768 | 2572 |
from t1M have "\<exists>(t1n, t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" |
2573 |
by auto |
|
2574 |
then obtain t1u t1n where t1uU: "(t1n, t1u) \<in> ?U" |
|
2575 |
and t1u: "t1 = - ?Nt a t1u / ?N t1n" |
|
2576 |
by blast |
|
2577 |
from t2M have "\<exists>(t2n, t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" |
|
2578 |
by auto |
|
2579 |
then obtain t2u t2n where t2uU: "(t2n, t2u) \<in> ?U" |
|
2580 |
and t2u: "t2 = - ?Nt a t2u / ?N t2n" |
|
2581 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2582 |
from t1x xt2 have t1t2: "t1 < t2" by simp |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2583 |
let ?u = "(t1 + t2) / 2" |
55768 | 2584 |
from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" |
2585 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2586 |
from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . |
55768 | 2587 |
with t1uU t2uU t1u t2u have ?thesis by blast |
2588 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2589 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2590 |
qed |
55768 | 2591 |
then obtain l n s m |
2592 |
where lnU: "(n, l) \<in> ?U" |
|
2593 |
and smU:"(m,s) \<in> ?U" |
|
2594 |
and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / 2) p" |
|
2595 |
by blast |
|
2596 |
from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" |
|
2597 |
by auto |
|
55754 | 2598 |
from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2599 |
tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu |
55768 | 2600 |
have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / 2) p" |
2601 |
by simp |
|
2602 |
with lnU smU show ?thesis by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2603 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2604 |
|
55768 | 2605 |
(* The Ferrante - Rackoff Theorem *) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2606 |
|
55754 | 2607 |
theorem fr_eq: |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2608 |
assumes lp: "islin p" |
55768 | 2609 |
shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow> |
2610 |
(Ifm vs (x#bs) (minusinf p) \<or> |
|
2611 |
Ifm vs (x#bs) (plusinf p) \<or> |
|
2612 |
(\<exists>(n, t) \<in> set (uset p). \<exists>(m, s) \<in> set (uset p). |
|
2613 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) / 2)#bs) p))" |
|
55754 | 2614 |
(is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2615 |
proof |
55754 | 2616 |
assume px: "\<exists>x. ?I x p" |
55768 | 2617 |
have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" |
2618 |
by blast |
|
2619 |
moreover { |
|
2620 |
assume "?M \<or> ?P" |
|
2621 |
then have "?D" by blast |
|
2622 |
} |
|
2623 |
moreover { |
|
2624 |
assume nmi: "\<not> ?M" |
|
2625 |
and npi: "\<not> ?P" |
|
2626 |
from inf_uset[OF lp nmi npi] have ?F |
|
2627 |
using px by blast |
|
2628 |
then have ?D by blast |
|
2629 |
} |
|
2630 |
ultimately show ?D by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2631 |
next |
55768 | 2632 |
assume ?D |
2633 |
moreover { |
|
2634 |
assume m: ?M |
|
2635 |
from minusinf_ex[OF lp m] have ?E . |
|
2636 |
} |
|
2637 |
moreover { |
|
2638 |
assume p: ?P |
|
2639 |
from plusinf_ex[OF lp p] have ?E . |
|
2640 |
} |
|
2641 |
moreover { |
|
2642 |
assume f: ?F |
|
2643 |
then have ?E by blast |
|
2644 |
} |
|
2645 |
ultimately show ?E by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2646 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2647 |
|
55768 | 2648 |
|
2649 |
section {* First implementation : Naive by encoding all case splits locally *} |
|
2650 |
||
55754 | 2651 |
definition "msubsteq c t d s a r = |
2652 |
evaldjf (split conj) |
|
55768 | 2653 |
[(let cd = c *\<^sub>p d |
2654 |
in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2655 |
(conj (Eq (CP c)) (NEq (CP d)), Eq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2656 |
(conj (NEq (CP c)) (Eq (CP d)), Eq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2657 |
(conj (Eq (CP c)) (Eq (CP d)), Eq r)]" |
|
2658 |
||
2659 |
lemma msubsteq_nb: |
|
2660 |
assumes lp: "islin (Eq (CNP 0 a r))" |
|
2661 |
and t: "tmbound0 t" |
|
2662 |
and s: "tmbound0 s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2663 |
shows "bound0 (msubsteq c t d s a r)" |
55768 | 2664 |
proof - |
2665 |
have th: "\<forall>x \<in> set |
|
2666 |
[(let cd = c *\<^sub>p d |
|
2667 |
in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2668 |
(conj (Eq (CP c)) (NEq (CP d)), Eq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2669 |
(conj (NEq (CP c)) (Eq (CP d)), Eq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2670 |
(conj (Eq (CP c)) (Eq (CP d)), Eq r)]. bound0 (split conj x)" |
|
2671 |
using lp by (simp add: Let_def t s) |
|
2672 |
from evaldjf_bound0[OF th] show ?thesis |
|
2673 |
by (simp add: msubsteq_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2674 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2675 |
|
55768 | 2676 |
lemma msubsteq: |
2677 |
assumes lp: "islin (Eq (CNP 0 a r))" |
|
2678 |
shows "Ifm vs (x#bs) (msubsteq c t d s a r) = |
|
2679 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2)#bs) (Eq (CNP 0 a r))" |
|
2680 |
(is "?lhs = ?rhs") |
|
2681 |
proof - |
|
2682 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2683 |
let ?N = "\<lambda>p. Ipoly vs p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2684 |
let ?c = "?N c" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2685 |
let ?d = "?N d" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2686 |
let ?t = "?Nt x t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2687 |
let ?s = "?Nt x s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2688 |
let ?a = "?N a" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2689 |
let ?r = "?Nt x r" |
55768 | 2690 |
from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" |
2691 |
by simp_all |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2692 |
note r= tmbound0_I[OF lin(3), of vs _ bs x] |
55768 | 2693 |
have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)" |
2694 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2695 |
moreover |
55768 | 2696 |
{ |
2697 |
assume c: "?c = 0" |
|
2698 |
and d: "?d=0" |
|
2699 |
then have ?thesis |
|
2700 |
by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex) |
|
2701 |
} |
|
55754 | 2702 |
moreover |
55768 | 2703 |
{ |
2704 |
assume c: "?c = 0" |
|
2705 |
and d: "?d\<noteq>0" |
|
2706 |
from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)" |
|
2707 |
by simp |
|
2708 |
have "?rhs = Ifm vs (-?s / (2*?d) # bs) (Eq (CNP 0 a r))" |
|
2709 |
by (simp only: th) |
|
2710 |
also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r = 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2711 |
by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"]) |
55768 | 2712 |
also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (-?s / (2*?d)) + ?r) = 0" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2713 |
using d mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp |
55768 | 2714 |
also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2 * ?d * ?r= 0" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48562
diff
changeset
|
2715 |
by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left) |
55768 | 2716 |
also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r = 0" |
2717 |
using d by simp |
|
2718 |
finally have ?thesis |
|
2719 |
using c d |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
45499
diff
changeset
|
2720 |
by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubsteq_def Let_def evaldjf_ex) |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2721 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2722 |
moreover |
55768 | 2723 |
{ |
2724 |
assume c: "?c \<noteq> 0" |
|
2725 |
and d: "?d = 0" |
|
2726 |
from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2 * ?c)" |
|
2727 |
by simp |
|
2728 |
have "?rhs = Ifm vs (-?t / (2*?c) # bs) (Eq (CNP 0 a r))" |
|
2729 |
by (simp only: th) |
|
2730 |
also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r = 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2731 |
by (simp add: r[of "- (?t/ (2 * ?c))"]) |
55768 | 2732 |
also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (-?t / (2 * ?c)) + ?r) = 0" |
2733 |
using c mult_cancel_left[of "2 * ?c" "(?a * (-?t / (2 * ?c)) + ?r)" 0] by simp |
|
2734 |
also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2 * ?c) / (2 * ?c) + 2 * ?c * ?r= 0" |
|
2735 |
by (simp add: field_simps distrib_left[of "2 * ?c"] del: distrib_left) |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2736 |
also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2 * ?c * ?r = 0" using c by simp |
55754 | 2737 |
finally have ?thesis using c d |
55768 | 2738 |
by (simp add: r[of "- (?t/ (2 * ?c))"] msubsteq_def Let_def evaldjf_ex) |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2739 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2740 |
moreover |
55768 | 2741 |
{ |
2742 |
assume c: "?c \<noteq> 0" |
|
2743 |
and d: "?d \<noteq> 0" |
|
2744 |
then have dc: "?c * ?d * 2 \<noteq> 0" |
|
2745 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2746 |
from add_frac_eq[OF c d, of "- ?t" "- ?s"] |
55768 | 2747 |
have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
2748 |
by (simp add: field_simps) |
55768 | 2749 |
have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Eq (CNP 0 a r))" |
2750 |
by (simp only: th) |
|
55754 | 2751 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
2752 |
by (simp add: r [of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) |
55768 | 2753 |
also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) = 0" |
2754 |
using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2 * ?c * ?d)) + ?r" 0] |
|
2755 |
by simp |
|
2756 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2 * ?c * ?d * ?r = 0" |
|
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
2757 |
using nonzero_mult_divide_cancel_left [OF dc] c d |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2758 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55754 | 2759 |
finally have ?thesis using c d |
55768 | 2760 |
by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] |
2761 |
msubsteq_def Let_def evaldjf_ex field_simps) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2762 |
} |
55768 | 2763 |
ultimately show ?thesis |
2764 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2765 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2766 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2767 |
|
55754 | 2768 |
definition "msubstneq c t d s a r = |
2769 |
evaldjf (split conj) |
|
55768 | 2770 |
[(let cd = c *\<^sub>p d |
2771 |
in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2772 |
(conj (Eq (CP c)) (NEq (CP d)), NEq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2773 |
(conj (NEq (CP c)) (Eq (CP d)), NEq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2774 |
(conj (Eq (CP c)) (Eq (CP d)), NEq r)]" |
|
2775 |
||
2776 |
lemma msubstneq_nb: |
|
2777 |
assumes lp: "islin (NEq (CNP 0 a r))" |
|
2778 |
and t: "tmbound0 t" |
|
2779 |
and s: "tmbound0 s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2780 |
shows "bound0 (msubstneq c t d s a r)" |
55768 | 2781 |
proof - |
2782 |
have th: "\<forall>x\<in> set |
|
2783 |
[(let cd = c *\<^sub>p d |
|
2784 |
in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2785 |
(conj (Eq (CP c)) (NEq (CP d)), NEq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2786 |
(conj (NEq (CP c)) (Eq (CP d)), NEq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2787 |
(conj (Eq (CP c)) (Eq (CP d)), NEq r)]. bound0 (split conj x)" |
|
2788 |
using lp by (simp add: Let_def t s) |
|
2789 |
from evaldjf_bound0[OF th] show ?thesis |
|
2790 |
by (simp add: msubstneq_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2791 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2792 |
|
55768 | 2793 |
lemma msubstneq: |
2794 |
assumes lp: "islin (Eq (CNP 0 a r))" |
|
2795 |
shows "Ifm vs (x#bs) (msubstneq c t d s a r) = |
|
2796 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (NEq (CNP 0 a r))" |
|
2797 |
(is "?lhs = ?rhs") |
|
2798 |
proof - |
|
2799 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2800 |
let ?N = "\<lambda>p. Ipoly vs p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2801 |
let ?c = "?N c" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2802 |
let ?d = "?N d" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2803 |
let ?t = "?Nt x t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2804 |
let ?s = "?Nt x s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2805 |
let ?a = "?N a" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2806 |
let ?r = "?Nt x r" |
55768 | 2807 |
from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" |
2808 |
by simp_all |
|
2809 |
note r = tmbound0_I[OF lin(3), of vs _ bs x] |
|
2810 |
have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)" |
|
2811 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2812 |
moreover |
55768 | 2813 |
{ |
2814 |
assume c: "?c = 0" |
|
2815 |
and d: "?d=0" |
|
2816 |
then have ?thesis |
|
2817 |
by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex) |
|
2818 |
} |
|
55754 | 2819 |
moreover |
55768 | 2820 |
{ |
2821 |
assume c: "?c = 0" |
|
2822 |
and d: "?d\<noteq>0" |
|
2823 |
from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2 * ?d)" |
|
2824 |
by simp |
|
2825 |
have "?rhs = Ifm vs (-?s / (2*?d) # bs) (NEq (CNP 0 a r))" |
|
2826 |
by (simp only: th) |
|
2827 |
also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r \<noteq> 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2828 |
by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"]) |
55754 | 2829 |
also have "\<dots> \<longleftrightarrow> 2*?d * (?a * (-?s / (2*?d)) + ?r) \<noteq> 0" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2830 |
using d mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp |
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2831 |
also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2*?d*?r\<noteq> 0" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48562
diff
changeset
|
2832 |
by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left) |
55768 | 2833 |
also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r \<noteq> 0" |
2834 |
using d by simp |
|
2835 |
finally have ?thesis |
|
2836 |
using c d |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
45499
diff
changeset
|
2837 |
by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubstneq_def Let_def evaldjf_ex) |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2838 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2839 |
moreover |
55768 | 2840 |
{ |
2841 |
assume c: "?c \<noteq> 0" |
|
2842 |
and d: "?d=0" |
|
2843 |
from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)" |
|
2844 |
by simp |
|
2845 |
have "?rhs = Ifm vs (-?t / (2*?c) # bs) (NEq (CNP 0 a r))" |
|
2846 |
by (simp only: th) |
|
2847 |
also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r \<noteq> 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2848 |
by (simp add: r[of "- (?t/ (2 * ?c))"]) |
55754 | 2849 |
also have "\<dots> \<longleftrightarrow> 2*?c * (?a * (-?t / (2*?c)) + ?r) \<noteq> 0" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2850 |
using c mult_cancel_left[of "2*?c" "(?a * (-?t / (2*?c)) + ?r)" 0] by simp |
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2851 |
also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2*?c) / (2*?c) + 2*?c*?r \<noteq> 0" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48562
diff
changeset
|
2852 |
by (simp add: field_simps distrib_left[of "2*?c"] del: distrib_left) |
55768 | 2853 |
also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2*?c*?r \<noteq> 0" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2854 |
using c by simp |
55768 | 2855 |
finally have ?thesis |
2856 |
using c d by (simp add: r[of "- (?t/ (2*?c))"] msubstneq_def Let_def evaldjf_ex) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2857 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2858 |
moreover |
55768 | 2859 |
{ |
2860 |
assume c: "?c \<noteq> 0" |
|
2861 |
and d: "?d \<noteq> 0" |
|
2862 |
then have dc: "?c * ?d *2 \<noteq> 0" |
|
2863 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2864 |
from add_frac_eq[OF c d, of "- ?t" "- ?s"] |
55768 | 2865 |
have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)" |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
2866 |
by (simp add: field_simps) |
55768 | 2867 |
have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (NEq (CNP 0 a r))" |
2868 |
by (simp only: th) |
|
55754 | 2869 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<noteq> 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
2870 |
by (simp add: r [of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) |
55768 | 2871 |
also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<noteq> 0" |
2872 |
using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] |
|
2873 |
by simp |
|
55754 | 2874 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r \<noteq> 0" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2875 |
using nonzero_mult_divide_cancel_left[OF dc] c d |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2876 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55768 | 2877 |
finally have ?thesis |
2878 |
using c d |
|
2879 |
by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] |
|
2880 |
msubstneq_def Let_def evaldjf_ex field_simps) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2881 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2882 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2883 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2884 |
|
55754 | 2885 |
definition "msubstlt c t d s a r = |
2886 |
evaldjf (split conj) |
|
55768 | 2887 |
[(let cd = c *\<^sub>p d |
2888 |
in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2889 |
(let cd = c *\<^sub>p d |
|
2890 |
in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2891 |
(conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Lt (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2892 |
(conj (lt (CP c)) (Eq (CP d)), Lt (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2893 |
(conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Lt (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2894 |
(conj (lt (CP d)) (Eq (CP c)), Lt (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2895 |
(conj (Eq (CP c)) (Eq (CP d)), Lt r)]" |
|
2896 |
||
2897 |
lemma msubstlt_nb: |
|
2898 |
assumes lp: "islin (Lt (CNP 0 a r))" |
|
2899 |
and t: "tmbound0 t" |
|
2900 |
and s: "tmbound0 s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2901 |
shows "bound0 (msubstlt c t d s a r)" |
55768 | 2902 |
proof - |
2903 |
have th: "\<forall>x\<in> set |
|
2904 |
[(let cd = c *\<^sub>p d |
|
2905 |
in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2906 |
(let cd = c *\<^sub>p d |
|
2907 |
in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
2908 |
(conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Lt (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2909 |
(conj (lt (CP c)) (Eq (CP d)), Lt (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
2910 |
(conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Lt (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2911 |
(conj (lt (CP d)) (Eq (CP c)), Lt (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
2912 |
(conj (Eq (CP c)) (Eq (CP d)), Lt r)]. bound0 (split conj x)" |
|
2913 |
using lp by (simp add: Let_def t s lt_nb) |
|
2914 |
from evaldjf_bound0[OF th] show ?thesis |
|
2915 |
by (simp add: msubstlt_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2916 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2917 |
|
55768 | 2918 |
lemma msubstlt: |
2919 |
assumes nc: "isnpoly c" |
|
2920 |
and nd: "isnpoly d" |
|
2921 |
and lp: "islin (Lt (CNP 0 a r))" |
|
55754 | 2922 |
shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow> |
55768 | 2923 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Lt (CNP 0 a r))" |
2924 |
(is "?lhs = ?rhs") |
|
2925 |
proof - |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2926 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2927 |
let ?N = "\<lambda>p. Ipoly vs p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2928 |
let ?c = "?N c" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2929 |
let ?d = "?N d" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2930 |
let ?t = "?Nt x t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2931 |
let ?s = "?Nt x s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2932 |
let ?a = "?N a" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2933 |
let ?r = "?Nt x r" |
55768 | 2934 |
from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" |
2935 |
by simp_all |
|
2936 |
note r = tmbound0_I[OF lin(3), of vs _ bs x] |
|
2937 |
have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)" |
|
2938 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2939 |
moreover |
55768 | 2940 |
{ |
2941 |
assume c: "?c=0" and d: "?d=0" |
|
2942 |
then have ?thesis |
|
2943 |
using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex) |
|
2944 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2945 |
moreover |
55768 | 2946 |
{ |
2947 |
assume dc: "?c*?d > 0" |
|
2948 |
from dc have dc': "2*?c *?d > 0" |
|
2949 |
by simp |
|
2950 |
then have c:"?c \<noteq> 0" and d: "?d \<noteq> 0" |
|
2951 |
by auto |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2952 |
from dc' have dc'': "\<not> 2*?c *?d < 0" by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2953 |
from add_frac_eq[OF c d, of "- ?t" "- ?s"] |
55768 | 2954 |
have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
2955 |
by (simp add: field_simps) |
55768 | 2956 |
have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))" |
2957 |
by (simp only: th) |
|
55754 | 2958 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r < 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
2959 |
by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2960 |
also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) < 0" |
55768 | 2961 |
using dc' dc'' |
2962 |
mult_less_cancel_left_disj[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] |
|
2963 |
by simp |
|
55754 | 2964 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r < 0" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2965 |
using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2966 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2967 |
finally have ?thesis using dc c d nc nd dc' |
55768 | 2968 |
by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] |
2969 |
msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2970 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2971 |
moreover |
55768 | 2972 |
{ |
2973 |
assume dc: "?c * ?d < 0" |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2974 |
from dc have dc': "2*?c *?d < 0" |
55754 | 2975 |
by (simp add: mult_less_0_iff field_simps) |
55768 | 2976 |
then have c:"?c \<noteq> 0" and d: "?d \<noteq> 0" |
2977 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2978 |
from add_frac_eq[OF c d, of "- ?t" "- ?s"] |
55768 | 2979 |
have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
2980 |
by (simp add: field_simps) |
55768 | 2981 |
have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2 * ?c * ?d) # bs) (Lt (CNP 0 a r))" |
2982 |
by (simp only: th) |
|
2983 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)) + ?r < 0" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
2984 |
by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) |
55768 | 2985 |
also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)) + ?r) > 0" |
2986 |
using dc' order_less_not_sym[OF dc'] |
|
2987 |
mult_less_cancel_left_disj[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] |
|
2988 |
by simp |
|
2989 |
also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2 * ?c * ?d * ?r < 0" |
|
2990 |
using nonzero_mult_divide_cancel_left[of "2 * ?c * ?d"] c d |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2991 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55768 | 2992 |
finally have ?thesis using dc c d nc nd |
2993 |
by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] |
|
2994 |
msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
2995 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
2996 |
moreover |
55768 | 2997 |
{ |
2998 |
assume c: "?c > 0" and d: "?d = 0" |
|
2999 |
from c have c'': "2*?c > 0" |
|
3000 |
by (simp add: zero_less_mult_iff) |
|
3001 |
from c have c': "2 * ?c \<noteq> 0" |
|
3002 |
by simp |
|
3003 |
from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2 * ?c)" |
|
3004 |
by (simp add: field_simps) |
|
3005 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2 * ?c) # bs) (Lt (CNP 0 a r))" |
|
3006 |
by (simp only: th) |
|
3007 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2 * ?c))+ ?r < 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3008 |
by (simp add: r[of "- (?t / (2 * ?c))"]) |
55768 | 3009 |
also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) < 0" |
3010 |
using c mult_less_cancel_left_disj[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' |
|
3011 |
order_less_not_sym[OF c''] |
|
3012 |
by simp |
|
3013 |
also have "\<dots> \<longleftrightarrow> - ?a * ?t + 2 * ?c * ?r < 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3014 |
using nonzero_mult_divide_cancel_left[OF c'] c |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3015 |
by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) |
55754 | 3016 |
finally have ?thesis using c d nc nd |
55768 | 3017 |
by (simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps |
3018 |
lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3019 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3020 |
moreover |
55768 | 3021 |
{ |
3022 |
assume c: "?c < 0" and d: "?d = 0" |
|
3023 |
then have c': "2 * ?c \<noteq> 0" |
|
3024 |
by simp |
|
3025 |
from c have c'': "2 * ?c < 0" |
|
3026 |
by (simp add: mult_less_0_iff) |
|
3027 |
from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2 * ?c)" |
|
3028 |
by (simp add: field_simps) |
|
3029 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))" |
|
3030 |
by (simp only: th) |
|
3031 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2*?c))+ ?r < 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3032 |
by (simp add: r[of "- (?t / (2*?c))"]) |
55768 | 3033 |
also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) > 0" |
3034 |
using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' |
|
3035 |
mult_less_cancel_left_disj[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] |
|
3036 |
by simp |
|
55754 | 3037 |
also have "\<dots> \<longleftrightarrow> ?a*?t - 2*?c *?r < 0" |
55768 | 3038 |
using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] |
3039 |
less_imp_neq[OF c''] c'' |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3040 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55754 | 3041 |
finally have ?thesis using c d nc nd |
55768 | 3042 |
by (simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps |
3043 |
lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3044 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3045 |
moreover |
55768 | 3046 |
{ |
3047 |
assume c: "?c = 0" and d: "?d > 0" |
|
3048 |
from d have d'': "2 * ?d > 0" |
|
3049 |
by (simp add: zero_less_mult_iff) |
|
3050 |
from d have d': "2 * ?d \<noteq> 0" |
|
3051 |
by simp |
|
3052 |
from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2 * ?d)" |
|
3053 |
by (simp add: field_simps) |
|
3054 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2 * ?d) # bs) (Lt (CNP 0 a r))" |
|
3055 |
by (simp only: th) |
|
3056 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?s / (2 * ?d))+ ?r < 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3057 |
by (simp add: r[of "- (?s / (2 * ?d))"]) |
55768 | 3058 |
also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (- ?s / (2 * ?d))+ ?r) < 0" |
3059 |
using d mult_less_cancel_left_disj[of "2 * ?d" "?a * (- ?s / (2 * ?d))+ ?r" 0] d' d'' |
|
3060 |
order_less_not_sym[OF d''] |
|
3061 |
by simp |
|
3062 |
also have "\<dots> \<longleftrightarrow> - ?a * ?s+ 2 * ?d * ?r < 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3063 |
using nonzero_mult_divide_cancel_left[OF d'] d |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3064 |
by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) |
55754 | 3065 |
finally have ?thesis using c d nc nd |
55768 | 3066 |
by (simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps |
3067 |
lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3068 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3069 |
moreover |
55768 | 3070 |
{ |
3071 |
assume c: "?c = 0" and d: "?d < 0" |
|
3072 |
then have d': "2 * ?d \<noteq> 0" |
|
3073 |
by simp |
|
3074 |
from d have d'': "2 * ?d < 0" |
|
3075 |
by (simp add: mult_less_0_iff) |
|
3076 |
from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)" |
|
3077 |
by (simp add: field_simps) |
|
3078 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2 * ?d) # bs) (Lt (CNP 0 a r))" |
|
3079 |
by (simp only: th) |
|
3080 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?s / (2 * ?d)) + ?r < 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3081 |
by (simp add: r[of "- (?s / (2 * ?d))"]) |
55768 | 3082 |
also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (- ?s / (2 * ?d)) + ?r) > 0" |
3083 |
using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' |
|
3084 |
mult_less_cancel_left_disj[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] |
|
3085 |
by simp |
|
3086 |
also have "\<dots> \<longleftrightarrow> ?a * ?s - 2 * ?d * ?r < 0" |
|
3087 |
using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] |
|
3088 |
less_imp_neq[OF d''] d'' |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3089 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55754 | 3090 |
finally have ?thesis using c d nc nd |
55768 | 3091 |
by (simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps |
3092 |
lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3093 |
} |
55768 | 3094 |
ultimately show ?thesis by blast |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3095 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3096 |
|
55754 | 3097 |
definition "msubstle c t d s a r = |
3098 |
evaldjf (split conj) |
|
55768 | 3099 |
[(let cd = c *\<^sub>p d |
3100 |
in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
3101 |
(let cd = c *\<^sub>p d |
|
3102 |
in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
3103 |
(conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Le (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
3104 |
(conj (lt (CP c)) (Eq (CP d)), Le (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
3105 |
(conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Le (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
3106 |
(conj (lt (CP d)) (Eq (CP c)), Le (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
3107 |
(conj (Eq (CP c)) (Eq (CP d)), Le r)]" |
|
3108 |
||
3109 |
lemma msubstle_nb: |
|
3110 |
assumes lp: "islin (Le (CNP 0 a r))" |
|
3111 |
and t: "tmbound0 t" |
|
3112 |
and s: "tmbound0 s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3113 |
shows "bound0 (msubstle c t d s a r)" |
55768 | 3114 |
proof - |
3115 |
have th: "\<forall>x\<in> set |
|
3116 |
[(let cd = c *\<^sub>p d |
|
3117 |
in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
3118 |
(let cd = c *\<^sub>p d |
|
3119 |
in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))), |
|
3120 |
(conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
3121 |
(conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))), |
|
3122 |
(conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
3123 |
(conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))), |
|
3124 |
(conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)" |
|
3125 |
using lp by (simp add: Let_def t s lt_nb) |
|
3126 |
from evaldjf_bound0[OF th] show ?thesis |
|
3127 |
by (simp add: msubstle_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3128 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3129 |
|
55768 | 3130 |
lemma msubstle: |
3131 |
assumes nc: "isnpoly c" |
|
3132 |
and nd: "isnpoly d" |
|
3133 |
and lp: "islin (Le (CNP 0 a r))" |
|
55754 | 3134 |
shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow> |
55768 | 3135 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Le (CNP 0 a r))" |
3136 |
(is "?lhs = ?rhs") |
|
3137 |
proof - |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3138 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3139 |
let ?N = "\<lambda>p. Ipoly vs p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3140 |
let ?c = "?N c" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3141 |
let ?d = "?N d" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3142 |
let ?t = "?Nt x t" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3143 |
let ?s = "?Nt x s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3144 |
let ?a = "?N a" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3145 |
let ?r = "?Nt x r" |
55768 | 3146 |
from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" |
3147 |
by simp_all |
|
3148 |
note r = tmbound0_I[OF lin(3), of vs _ bs x] |
|
3149 |
have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)" |
|
3150 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3151 |
moreover |
55768 | 3152 |
{ |
3153 |
assume c: "?c = 0" and d: "?d = 0" |
|
3154 |
then have ?thesis |
|
3155 |
using nc nd |
|
3156 |
by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex) |
|
3157 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3158 |
moreover |
55768 | 3159 |
{ |
3160 |
assume dc: "?c * ?d > 0" |
|
3161 |
from dc have dc': "2 * ?c * ?d > 0" |
|
3162 |
by simp |
|
3163 |
then have c: "?c \<noteq> 0" and d: "?d \<noteq> 0" |
|
3164 |
by auto |
|
3165 |
from dc' have dc'': "\<not> 2 * ?c * ?d < 0" |
|
3166 |
by simp |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3167 |
from add_frac_eq[OF c d, of "- ?t" "- ?s"] |
55768 | 3168 |
have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)" |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
3169 |
by (simp add: field_simps) |
55768 | 3170 |
have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" |
3171 |
by (simp only: th) |
|
3172 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<le> 0" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3173 |
by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) |
55768 | 3174 |
also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<le> 0" |
3175 |
using dc' dc'' |
|
3176 |
mult_le_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] |
|
3177 |
by simp |
|
3178 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r \<le> 0" |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3179 |
using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3180 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3181 |
finally have ?thesis using dc c d nc nd dc' |
55768 | 3182 |
by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def |
3183 |
Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3184 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3185 |
moreover |
55768 | 3186 |
{ |
3187 |
assume dc: "?c * ?d < 0" |
|
3188 |
from dc have dc': "2 * ?c * ?d < 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3189 |
by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos) |
55768 | 3190 |
then have c: "?c \<noteq> 0" and d: "?d \<noteq> 0" |
3191 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3192 |
from add_frac_eq[OF c d, of "- ?t" "- ?s"] |
55768 | 3193 |
have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
3194 |
by (simp add: field_simps) |
55768 | 3195 |
have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" |
3196 |
by (simp only: th) |
|
3197 |
also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<le> 0" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3198 |
by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) |
55768 | 3199 |
also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<ge> 0" |
3200 |
using dc' order_less_not_sym[OF dc'] |
|
3201 |
mult_le_cancel_left[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] |
|
3202 |
by simp |
|
3203 |
also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2 * ?c * ?d * ?r \<le> 0" |
|
3204 |
using nonzero_mult_divide_cancel_left[of "2 * ?c * ?d"] c d |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3205 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3206 |
finally have ?thesis using dc c d nc nd |
55768 | 3207 |
by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def |
3208 |
Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3209 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3210 |
moreover |
55768 | 3211 |
{ |
3212 |
assume c: "?c > 0" and d: "?d = 0" |
|
3213 |
from c have c'': "2 * ?c > 0" |
|
3214 |
by (simp add: zero_less_mult_iff) |
|
3215 |
from c have c': "2 * ?c \<noteq> 0" |
|
3216 |
by simp |
|
3217 |
from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)" |
|
3218 |
by (simp add: field_simps) |
|
3219 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2 * ?c) # bs) (Le (CNP 0 a r))" |
|
3220 |
by (simp only: th) |
|
3221 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2 * ?c))+ ?r \<le> 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3222 |
by (simp add: r[of "- (?t / (2 * ?c))"]) |
55768 | 3223 |
also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) \<le> 0" |
3224 |
using c mult_le_cancel_left[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' |
|
3225 |
order_less_not_sym[OF c''] |
|
3226 |
by simp |
|
3227 |
also have "\<dots> \<longleftrightarrow> - ?a*?t+ 2*?c *?r \<le> 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3228 |
using nonzero_mult_divide_cancel_left[OF c'] c |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3229 |
by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) |
55754 | 3230 |
finally have ?thesis using c d nc nd |
55768 | 3231 |
by (simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def |
3232 |
evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3233 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3234 |
moreover |
55768 | 3235 |
{ |
3236 |
assume c: "?c < 0" and d: "?d = 0" |
|
3237 |
then have c': "2 * ?c \<noteq> 0" |
|
3238 |
by simp |
|
3239 |
from c have c'': "2 * ?c < 0" |
|
3240 |
by (simp add: mult_less_0_iff) |
|
3241 |
from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)" |
|
3242 |
by (simp add: field_simps) |
|
3243 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2 * ?c) # bs) (Le (CNP 0 a r))" |
|
3244 |
by (simp only: th) |
|
3245 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2*?c))+ ?r \<le> 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3246 |
by (simp add: r[of "- (?t / (2*?c))"]) |
55768 | 3247 |
also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) \<ge> 0" |
3248 |
using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' |
|
3249 |
mult_le_cancel_left[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] |
|
3250 |
by simp |
|
3251 |
also have "\<dots> \<longleftrightarrow> ?a * ?t - 2 * ?c * ?r \<le> 0" |
|
3252 |
using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] |
|
3253 |
less_imp_neq[OF c''] c'' |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3254 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55754 | 3255 |
finally have ?thesis using c d nc nd |
55768 | 3256 |
by (simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def |
3257 |
evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3258 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3259 |
moreover |
55768 | 3260 |
{ |
3261 |
assume c: "?c = 0" and d: "?d > 0" |
|
3262 |
from d have d'': "2 * ?d > 0" |
|
3263 |
by (simp add: zero_less_mult_iff) |
|
3264 |
from d have d': "2 * ?d \<noteq> 0" |
|
3265 |
by simp |
|
3266 |
from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2 * ?d)" |
|
3267 |
by (simp add: field_simps) |
|
3268 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2 * ?d) # bs) (Le (CNP 0 a r))" |
|
3269 |
by (simp only: th) |
|
3270 |
also have "\<dots> \<longleftrightarrow> ?a * (- ?s / (2 * ?d))+ ?r \<le> 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3271 |
by (simp add: r[of "- (?s / (2*?d))"]) |
55768 | 3272 |
also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (- ?s / (2 * ?d)) + ?r) \<le> 0" |
3273 |
using d mult_le_cancel_left[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d'' |
|
3274 |
order_less_not_sym[OF d''] |
|
3275 |
by simp |
|
3276 |
also have "\<dots> \<longleftrightarrow> - ?a * ?s + 2 * ?d * ?r \<le> 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3277 |
using nonzero_mult_divide_cancel_left[OF d'] d |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3278 |
by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) |
55754 | 3279 |
finally have ?thesis using c d nc nd |
55768 | 3280 |
by (simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def |
3281 |
evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3282 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3283 |
moreover |
55768 | 3284 |
{ |
3285 |
assume c: "?c = 0" and d: "?d < 0" |
|
3286 |
then have d': "2 * ?d \<noteq> 0" |
|
3287 |
by simp |
|
3288 |
from d have d'': "2 * ?d < 0" |
|
3289 |
by (simp add: mult_less_0_iff) |
|
3290 |
from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)" |
|
3291 |
by (simp add: field_simps) |
|
3292 |
have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))" |
|
3293 |
by (simp only: th) |
|
3294 |
also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r \<le> 0" |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3295 |
by (simp add: r[of "- (?s / (2*?d))"]) |
55768 | 3296 |
also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) \<ge> 0" |
3297 |
using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' |
|
3298 |
mult_le_cancel_left[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] |
|
3299 |
by simp |
|
3300 |
also have "\<dots> \<longleftrightarrow> ?a * ?s - 2 * ?d * ?r \<le> 0" |
|
3301 |
using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] |
|
3302 |
less_imp_neq[OF d''] d'' |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3303 |
by (simp add: algebra_simps diff_divide_distrib del: distrib_right) |
55754 | 3304 |
finally have ?thesis using c d nc nd |
55768 | 3305 |
by (simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def |
3306 |
evaldjf_ex field_simps lt polyneg_norm polymul_norm) |
|
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3307 |
} |
55768 | 3308 |
ultimately show ?thesis by blast |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3309 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3310 |
|
55768 | 3311 |
fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" |
3312 |
where |
|
3313 |
"msubst (And p q) ((c, t), (d, s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c, t), (d, s)))" |
|
3314 |
| "msubst (Or p q) ((c, t), (d, s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c, t), (d, s)))" |
|
3315 |
| "msubst (Eq (CNP 0 a r)) ((c, t), (d, s)) = msubsteq c t d s a r" |
|
3316 |
| "msubst (NEq (CNP 0 a r)) ((c, t), (d, s)) = msubstneq c t d s a r" |
|
3317 |
| "msubst (Lt (CNP 0 a r)) ((c, t), (d, s)) = msubstlt c t d s a r" |
|
3318 |
| "msubst (Le (CNP 0 a r)) ((c, t), (d, s)) = msubstle c t d s a r" |
|
3319 |
| "msubst p ((c, t), (d, s)) = p" |
|
3320 |
||
3321 |
lemma msubst_I: |
|
3322 |
assumes lp: "islin p" |
|
3323 |
and nc: "isnpoly c" |
|
3324 |
and nd: "isnpoly d" |
|
3325 |
shows "Ifm vs (x#bs) (msubst p ((c,t),(d,s))) = |
|
3326 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) p" |
|
3327 |
using lp |
|
3328 |
by (induct p rule: islin.induct) |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3329 |
(auto simp add: tmbound0_I |
55768 | 3330 |
[where b = "(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) / 2" |
3331 |
and b' = x and bs = bs and vs = vs] |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3332 |
msubsteq msubstneq msubstlt [OF nc nd] msubstle [OF nc nd]) |
55768 | 3333 |
|
3334 |
lemma msubst_nb: |
|
3335 |
assumes lp: "islin p" |
|
3336 |
and t: "tmbound0 t" |
|
3337 |
and s: "tmbound0 s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3338 |
shows "bound0 (msubst p ((c,t),(d,s)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3339 |
using lp t s |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3340 |
by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3341 |
|
55754 | 3342 |
lemma fr_eq_msubst: |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3343 |
assumes lp: "islin p" |
55768 | 3344 |
shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow> |
3345 |
(Ifm vs (x#bs) (minusinf p) \<or> |
|
3346 |
Ifm vs (x#bs) (plusinf p) \<or> |
|
3347 |
(\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p). |
|
3348 |
Ifm vs (x#bs) (msubst p ((c, t), (d, s)))))" |
|
55754 | 3349 |
(is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") |
55768 | 3350 |
proof - |
3351 |
from uset_l[OF lp] |
|
3352 |
have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" |
|
3353 |
by blast |
|
3354 |
{ |
|
3355 |
fix c t d s |
|
3356 |
assume ctU: "(c, t) \<in>set (uset p)" |
|
3357 |
and dsU: "(d,s) \<in>set (uset p)" |
|
3358 |
and pts: "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p" |
|
3359 |
from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" |
|
3360 |
by simp_all |
|
3361 |
from msubst_I[OF lp norm, of vs x bs t s] pts |
|
3362 |
have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" .. |
|
3363 |
} |
|
3364 |
moreover |
|
3365 |
{ |
|
3366 |
fix c t d s |
|
3367 |
assume ctU: "(c, t) \<in> set (uset p)" |
|
3368 |
and dsU: "(d,s) \<in>set (uset p)" |
|
3369 |
and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))" |
|
3370 |
from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" |
|
3371 |
by simp_all |
|
3372 |
from msubst_I[OF lp norm, of vs x bs t s] pts |
|
3373 |
have "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p" .. |
|
3374 |
} |
|
3375 |
ultimately have th': "(\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p). |
|
3376 |
Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p) \<longleftrightarrow> ?F" |
|
3377 |
by blast |
|
3378 |
from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis . |
|
55754 | 3379 |
qed |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3380 |
|
55768 | 3381 |
lemma simpfm_lin: |
3382 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3383 |
shows "qfree p \<Longrightarrow> islin (simpfm p)" |
55768 | 3384 |
by (induct p rule: simpfm.induct) (auto simp add: conj_lin disj_lin) |
3385 |
||
3386 |
definition "ferrack p \<equiv> |
|
3387 |
let |
|
3388 |
q = simpfm p; |
|
3389 |
mp = minusinf q; |
|
3390 |
pp = plusinf q |
|
3391 |
in |
|
3392 |
if (mp = T \<or> pp = T) then T |
|
3393 |
else |
|
3394 |
(let U = alluopairs (remdups (uset q)) |
|
3395 |
in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3396 |
|
55754 | 3397 |
lemma ferrack: |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3398 |
assumes qf: "qfree p" |
55768 | 3399 |
shows "qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)" |
3400 |
(is "_ \<and> ?rhs = ?lhs") |
|
3401 |
proof - |
|
3402 |
let ?I = "\<lambda>x p. Ifm vs (x#bs) p" |
|
3403 |
let ?N = "\<lambda>t. Ipoly vs t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3404 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
55754 | 3405 |
let ?q = "simpfm p" |
41823 | 3406 |
let ?U = "remdups(uset ?q)" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3407 |
let ?Up = "alluopairs ?U" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3408 |
let ?mp = "minusinf ?q" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3409 |
let ?pp = "plusinf ?q" |
55768 | 3410 |
fix x |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3411 |
let ?I = "\<lambda>p. Ifm vs (x#bs) p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3412 |
from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" . |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3413 |
from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" . |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3414 |
from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" . |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3415 |
from uset_l[OF lq] have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3416 |
by simp |
55768 | 3417 |
{ |
3418 |
fix c t d s |
|
3419 |
assume ctU: "(c, t) \<in> set ?U" |
|
3420 |
and dsU: "(d,s) \<in> set ?U" |
|
3421 |
from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" |
|
3422 |
by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3423 |
from msubst_I[OF lq norm, of vs x bs t s] msubst_I[OF lq norm(2,1), of vs x bs s t] |
55768 | 3424 |
have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" |
3425 |
by (simp add: field_simps) |
|
3426 |
} |
|
3427 |
then have th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (msubst ?q (x, y)) \<longleftrightarrow> ?I (msubst ?q (y, x))" |
|
3428 |
by auto |
|
3429 |
{ |
|
3430 |
fix x |
|
3431 |
assume xUp: "x \<in> set ?Up" |
|
3432 |
then obtain c t d s |
|
3433 |
where ctU: "(c, t) \<in> set ?U" |
|
3434 |
and dsU: "(d,s) \<in> set ?U" |
|
3435 |
and x: "x = ((c, t),(d, s))" |
|
3436 |
using alluopairs_set1[of ?U] by auto |
|
55754 | 3437 |
from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU] |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3438 |
have nbs: "tmbound0 t" "tmbound0 s" by simp_all |
55754 | 3439 |
from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]] |
55768 | 3440 |
have "bound0 ((simpfm o (msubst (simpfm p))) x)" |
3441 |
using x by simp |
|
3442 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3443 |
with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"] |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3444 |
have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast |
55754 | 3445 |
with mp_nb pp_nb |
55768 | 3446 |
have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" |
3447 |
by simp |
|
3448 |
from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" |
|
3449 |
by (simp add: ferrack_def Let_def) |
|
3450 |
have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" |
|
3451 |
by simp |
|
3452 |
also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> |
|
3453 |
(\<exists>(c, t)\<in>set ?U. \<exists>(d, s)\<in>set ?U. ?I (msubst (simpfm p) ((c, t), d, s)))" |
|
3454 |
using fr_eq_msubst[OF lq, of vs bs x] by simp |
|
3455 |
also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> |
|
3456 |
(\<exists>(x, y) \<in> set ?Up. ?I ((simpfm \<circ> msubst ?q) (x, y)))" |
|
3457 |
using alluopairs_bex[OF th0] by simp |
|
3458 |
also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm \<circ> msubst ?q) ?Up)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3459 |
by (simp add: evaldjf_ex) |
55768 | 3460 |
also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm \<circ> msubst ?q) ?Up)))" |
3461 |
by simp |
|
3462 |
also have "\<dots> \<longleftrightarrow> ?rhs" |
|
3463 |
using decr0[OF th1, of vs x bs] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3464 |
apply (simp add: ferrack_def Let_def) |
55768 | 3465 |
apply (cases "?mp = T \<or> ?pp = T") |
3466 |
apply auto |
|
3467 |
done |
|
3468 |
finally show ?thesis |
|
3469 |
using thqf by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3470 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3471 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3472 |
definition "frpar p = simpfm (qelim p ferrack)" |
55768 | 3473 |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3474 |
lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)" |
55768 | 3475 |
proof - |
3476 |
from ferrack |
|
3477 |
have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)" |
|
3478 |
by blast |
|
3479 |
from qelim[OF th, of p bs] show ?thesis |
|
3480 |
unfolding frpar_def by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3481 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3482 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3483 |
|
55768 | 3484 |
section {* Second implemenation: Case splits not local *} |
3485 |
||
3486 |
lemma fr_eq2: |
|
3487 |
assumes lp: "islin p" |
|
55754 | 3488 |
shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow> |
55768 | 3489 |
(Ifm vs (x#bs) (minusinf p) \<or> |
3490 |
Ifm vs (x#bs) (plusinf p) \<or> |
|
3491 |
Ifm vs (0#bs) p \<or> |
|
3492 |
(\<exists>(n, t) \<in> set (uset p). |
|
3493 |
Ipoly vs n \<noteq> 0 \<and> Ifm vs ((- Itm vs (x#bs) t / (Ipoly vs n * 2))#bs) p) \<or> |
|
3494 |
(\<exists>(n, t) \<in> set (uset p). \<exists>(m, s) \<in> set (uset p). |
|
3495 |
Ipoly vs n \<noteq> 0 \<and> |
|
3496 |
Ipoly vs m \<noteq> 0 \<and> |
|
3497 |
Ifm vs (((- Itm vs (x#bs) t / Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /2)#bs) p))" |
|
55754 | 3498 |
(is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D") |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3499 |
proof |
55754 | 3500 |
assume px: "\<exists>x. ?I x p" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3501 |
have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast |
55768 | 3502 |
moreover { |
3503 |
assume "?M \<or> ?P" |
|
3504 |
then have "?D" by blast |
|
3505 |
} |
|
3506 |
moreover { |
|
3507 |
assume nmi: "\<not> ?M" |
|
3508 |
and npi: "\<not> ?P" |
|
55754 | 3509 |
from inf_uset[OF lp nmi npi, OF px] |
55768 | 3510 |
obtain c t d s where ct: |
3511 |
"(c, t) \<in> set (uset p)" |
|
3512 |
"(d, s) \<in> set (uset p)" |
|
3513 |
"?I ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1)) p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3514 |
by auto |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3515 |
let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3516 |
let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3517 |
let ?s = "Itm vs (x # bs) s" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3518 |
let ?t = "Itm vs (x # bs) t" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3519 |
have eq2: "\<And>(x::'a). x + x = 2 * x" |
55768 | 3520 |
by (simp add: field_simps) |
3521 |
{ |
|
3522 |
assume "?c = 0 \<and> ?d = 0" |
|
3523 |
with ct have ?D by simp |
|
3524 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3525 |
moreover |
55768 | 3526 |
{ |
3527 |
assume z: "?c = 0" "?d \<noteq> 0" |
|
3528 |
from z have ?D using ct by auto |
|
3529 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3530 |
moreover |
55768 | 3531 |
{ |
3532 |
assume z: "?c \<noteq> 0" "?d = 0" |
|
3533 |
with ct have ?D by auto |
|
3534 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3535 |
moreover |
55768 | 3536 |
{ |
3537 |
assume z: "?c \<noteq> 0" "?d \<noteq> 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3538 |
from z have ?F using ct |
55768 | 3539 |
apply - |
3540 |
apply (rule bexI[where x = "(c,t)"]) |
|
3541 |
apply simp_all |
|
3542 |
apply (rule bexI[where x = "(d,s)"]) |
|
3543 |
apply simp_all |
|
3544 |
done |
|
3545 |
then have ?D by blast |
|
3546 |
} |
|
3547 |
ultimately have ?D by auto |
|
3548 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3549 |
ultimately show "?D" by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3550 |
next |
55754 | 3551 |
assume "?D" |
55768 | 3552 |
moreover { |
3553 |
assume m: "?M" |
|
3554 |
from minusinf_ex[OF lp m] have "?E" . |
|
3555 |
} |
|
3556 |
moreover { |
|
3557 |
assume p: "?P" |
|
3558 |
from plusinf_ex[OF lp p] have "?E" . |
|
3559 |
} |
|
3560 |
moreover { |
|
3561 |
assume f:"?F" |
|
3562 |
then have "?E" by blast |
|
3563 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3564 |
ultimately show "?E" by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3565 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3566 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3567 |
definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3568 |
definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3569 |
definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3570 |
definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3571 |
definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3572 |
|
55754 | 3573 |
lemma msubsteq2: |
55768 | 3574 |
assumes nz: "Ipoly vs c \<noteq> 0" |
3575 |
and l: "islin (Eq (CNP 0 a b))" |
|
3576 |
shows "Ifm vs (x#bs) (msubsteq2 c t a b) = |
|
3577 |
Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Eq (CNP 0 a b))" |
|
3578 |
using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3579 |
by (simp add: msubsteq2_def field_simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3580 |
|
55754 | 3581 |
lemma msubstltpos: |
55768 | 3582 |
assumes nz: "Ipoly vs c > 0" |
3583 |
and l: "islin (Lt (CNP 0 a b))" |
|
3584 |
shows "Ifm vs (x#bs) (msubstltpos c t a b) = |
|
3585 |
Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Lt (CNP 0 a b))" |
|
3586 |
using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3587 |
by (simp add: msubstltpos_def field_simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3588 |
|
55754 | 3589 |
lemma msubstlepos: |
55768 | 3590 |
assumes nz: "Ipoly vs c > 0" |
3591 |
and l: "islin (Le (CNP 0 a b))" |
|
3592 |
shows "Ifm vs (x#bs) (msubstlepos c t a b) = |
|
3593 |
Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Le (CNP 0 a b))" |
|
3594 |
using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3595 |
by (simp add: msubstlepos_def field_simps) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3596 |
|
55754 | 3597 |
lemma msubstltneg: |
55768 | 3598 |
assumes nz: "Ipoly vs c < 0" |
3599 |
and l: "islin (Lt (CNP 0 a b))" |
|
3600 |
shows "Ifm vs (x#bs) (msubstltneg c t a b) = |
|
3601 |
Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Lt (CNP 0 a b))" |
|
3602 |
using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3603 |
by (simp add: msubstltneg_def field_simps del: minus_add_distrib) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3604 |
|
55754 | 3605 |
lemma msubstleneg: |
55768 | 3606 |
assumes nz: "Ipoly vs c < 0" |
3607 |
and l: "islin (Le (CNP 0 a b))" |
|
3608 |
shows "Ifm vs (x#bs) (msubstleneg c t a b) = |
|
3609 |
Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Le (CNP 0 a b))" |
|
3610 |
using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric] |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3611 |
by (simp add: msubstleneg_def field_simps del: minus_add_distrib) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3612 |
|
55768 | 3613 |
fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" |
3614 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3615 |
"msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3616 |
| "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3617 |
| "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3618 |
| "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3619 |
| "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3620 |
| "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3621 |
| "msubstpos p c t = p" |
55754 | 3622 |
|
3623 |
lemma msubstpos_I: |
|
55768 | 3624 |
assumes lp: "islin p" |
3625 |
and pos: "Ipoly vs c > 0" |
|
3626 |
shows "Ifm vs (x#bs) (msubstpos p c t) = |
|
3627 |
Ifm vs (Itm vs (x#bs) t / Ipoly vs c #bs) p" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3628 |
using lp pos |
55768 | 3629 |
by (induct p rule: islin.induct) |
3630 |
(auto simp add: msubsteq2 msubstltpos[OF pos] msubstlepos[OF pos] |
|
3631 |
tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] |
|
3632 |
bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps) |
|
3633 |
||
3634 |
fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" |
|
3635 |
where |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3636 |
"msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3637 |
| "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3638 |
| "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3639 |
| "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3640 |
| "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3641 |
| "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3642 |
| "msubstneg p c t = p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3643 |
|
55754 | 3644 |
lemma msubstneg_I: |
55768 | 3645 |
assumes lp: "islin p" |
3646 |
and pos: "Ipoly vs c < 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3647 |
shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t / Ipoly vs c #bs) p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3648 |
using lp pos |
55768 | 3649 |
by (induct p rule: islin.induct) |
3650 |
(auto simp add: msubsteq2 msubstltneg[OF pos] msubstleneg[OF pos] |
|
3651 |
tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] |
|
3652 |
bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps) |
|
3653 |
||
3654 |
definition |
|
3655 |
"msubst2 p c t = |
|
3656 |
disj |
|
3657 |
(conj (lt (CP (polyneg c))) (simpfm (msubstpos p c t))) |
|
3658 |
(conj (lt (CP c)) (simpfm (msubstneg p c t)))" |
|
3659 |
||
3660 |
lemma msubst2: |
|
3661 |
assumes lp: "islin p" |
|
3662 |
and nc: "isnpoly c" |
|
3663 |
and nz: "Ipoly vs c \<noteq> 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3664 |
shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t / Ipoly vs c #bs) p" |
55768 | 3665 |
proof - |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3666 |
let ?c = "Ipoly vs c" |
55754 | 3667 |
from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3668 |
by (simp_all add: polyneg_norm) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3669 |
from nz have "?c > 0 \<or> ?c < 0" by arith |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3670 |
moreover |
55768 | 3671 |
{ |
3672 |
assume c: "?c < 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3673 |
from c msubstneg_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"] |
55768 | 3674 |
have ?thesis by (auto simp add: msubst2_def) |
3675 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3676 |
moreover |
55768 | 3677 |
{ |
3678 |
assume c: "?c > 0" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3679 |
from c msubstpos_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"] |
55768 | 3680 |
have ?thesis by (auto simp add: msubst2_def) |
3681 |
} |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3682 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3683 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3684 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3685 |
lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3686 |
by (simp add: msubsteq2_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3687 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3688 |
lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3689 |
by (simp add: msubstltpos_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3690 |
lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3691 |
by (simp add: msubstltneg_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3692 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3693 |
lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3694 |
by (simp add: msubstlepos_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3695 |
lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3696 |
by (simp add: msubstleneg_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3697 |
|
55768 | 3698 |
lemma msubstpos_nb: |
3699 |
assumes lp: "islin p" |
|
3700 |
and tnb: "tmbound0 t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3701 |
shows "bound0 (msubstpos p c t)" |
55768 | 3702 |
using lp tnb |
3703 |
by (induct p c t rule: msubstpos.induct) |
|
3704 |
(auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb) |
|
3705 |
||
3706 |
lemma msubstneg_nb: |
|
3707 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
3708 |
and lp: "islin p" |
|
3709 |
and tnb: "tmbound0 t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3710 |
shows "bound0 (msubstneg p c t)" |
55768 | 3711 |
using lp tnb |
3712 |
by (induct p c t rule: msubstneg.induct) |
|
3713 |
(auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb) |
|
3714 |
||
3715 |
lemma msubst2_nb: |
|
3716 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
|
3717 |
and lp: "islin p" |
|
3718 |
and tnb: "tmbound0 t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3719 |
shows "bound0 (msubst2 p c t)" |
55768 | 3720 |
using lp tnb |
3721 |
by (simp add: msubst2_def msubstneg_nb msubstpos_nb lt_nb simpfm_bound0) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3722 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
45499
diff
changeset
|
3723 |
lemma mult_minus2_left: "-2 * (x::'a::comm_ring_1) = - (2 * x)" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3724 |
by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3725 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
45499
diff
changeset
|
3726 |
lemma mult_minus2_right: "(x::'a::comm_ring_1) * -2 = - (x * 2)" |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3727 |
by simp |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3728 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3729 |
lemma islin_qf: "islin p \<Longrightarrow> qfree p" |
55768 | 3730 |
by (induct p rule: islin.induct) (auto simp add: bound0_qf) |
3731 |
||
55754 | 3732 |
lemma fr_eq_msubst2: |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3733 |
assumes lp: "islin p" |
55768 | 3734 |
shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow> |
3735 |
((Ifm vs (x#bs) (minusinf p)) \<or> |
|
3736 |
(Ifm vs (x#bs) (plusinf p)) \<or> |
|
3737 |
Ifm vs (x#bs) (subst0 (CP 0\<^sub>p) p) \<or> |
|
3738 |
(\<exists>(n, t) \<in> set (uset p). |
|
3739 |
Ifm vs (x# bs) (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<or> |
|
3740 |
(\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p). |
|
3741 |
Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))))" |
|
55754 | 3742 |
(is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D") |
55768 | 3743 |
proof - |
3744 |
from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" |
|
3745 |
by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3746 |
let ?I = "\<lambda>p. Ifm vs (x#bs) p" |
55768 | 3747 |
have n2: "isnpoly (C (-2,1))" |
3748 |
by (simp add: isnpoly_def) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3749 |
note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified] |
55754 | 3750 |
|
55768 | 3751 |
have eq1: "(\<exists>(n, t) \<in> set (uset p). ?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<longleftrightarrow> |
3752 |
(\<exists>(n, t) \<in> set (uset p). |
|
3753 |
\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> |
|
3754 |
Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p)" |
|
3755 |
proof - |
|
3756 |
{ |
|
3757 |
fix n t |
|
3758 |
assume H: "(n, t) \<in> set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)" |
|
3759 |
from H(1) th have "isnpoly n" |
|
3760 |
by blast |
|
3761 |
then have nn: "isnpoly (n *\<^sub>p (C (-2,1)))" |
|
3762 |
by (simp_all add: polymul_norm n2) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3763 |
have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
3764 |
by (simp add: polyneg_norm nn) |
55768 | 3765 |
then have nn2: "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>n \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
3766 |
using H(2) nn' nn |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
3767 |
by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3768 |
from msubst2[OF lp nn nn2(1), of x bs t] |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
3769 |
have "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3770 |
using H(2) nn2 by (simp add: mult_minus2_right) |
55768 | 3771 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3772 |
moreover |
55768 | 3773 |
{ |
3774 |
fix n t |
|
3775 |
assume H: |
|
3776 |
"(n, t) \<in> set (uset p)" "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
|
3777 |
"Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p" |
|
3778 |
from H(1) th have "isnpoly n" |
|
3779 |
by blast |
|
3780 |
then have nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
3781 |
using H(2) by (simp_all add: polymul_norm n2) |
55768 | 3782 |
from msubst2[OF lp nn, of x bs t] have "?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
3783 |
using H(2,3) by (simp add: mult_minus2_right) |
55768 | 3784 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3785 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3786 |
qed |
55768 | 3787 |
have eq2: "(\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p). |
3788 |
Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))) \<longleftrightarrow> |
|
3789 |
(\<exists>(n, t)\<in>set (uset p). \<exists>(m, s)\<in>set (uset p). |
|
3790 |
\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> |
|
3791 |
\<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> |
|
3792 |
Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p)" |
|
3793 |
proof - |
|
3794 |
{ |
|
3795 |
fix c t d s |
|
3796 |
assume H: |
|
3797 |
"(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" |
|
3798 |
"Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))" |
|
3799 |
from H(1,2) th have "isnpoly c" "isnpoly d" |
|
3800 |
by blast+ |
|
3801 |
then have nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
3802 |
by (simp_all add: polymul_norm n2) |
55768 | 3803 |
have stupid: |
3804 |
"allpolys isnpoly (CP (~\<^sub>p (C (-2, 1) *\<^sub>p c *\<^sub>p d)))" |
|
3805 |
"allpolys isnpoly (CP ((C (-2, 1) *\<^sub>p c *\<^sub>p d)))" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
3806 |
by (simp_all add: polyneg_norm nn) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3807 |
have nn': "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
55768 | 3808 |
using H(3) |
3809 |
by (auto simp add: msubst2_def lt[OF stupid(1)] |
|
3810 |
lt[OF stupid(2)] zero_less_mult_iff mult_less_0_iff) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3811 |
from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn' |
55768 | 3812 |
have "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> |
3813 |
Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
3814 |
by (simp add: add_divide_distrib diff_divide_distrib mult_minus2_left mult.commute) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3815 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3816 |
moreover |
55768 | 3817 |
{ |
3818 |
fix c t d s |
|
3819 |
assume H: |
|
3820 |
"(c, t) \<in> set (uset p)" |
|
3821 |
"(d, s) \<in> set (uset p)" |
|
3822 |
"\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
|
3823 |
"\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
|
3824 |
"Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p" |
|
3825 |
from H(1,2) th have "isnpoly c" "isnpoly d" |
|
3826 |
by blast+ |
|
3827 |
then have nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33212
diff
changeset
|
3828 |
using H(3,4) by (simp_all add: polymul_norm n2) |
55754 | 3829 |
from msubst2[OF lp nn, of x bs ] H(3,4,5) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3830 |
have "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56479
diff
changeset
|
3831 |
by (simp add: diff_divide_distrib add_divide_distrib mult_minus2_left mult.commute) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3832 |
} |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3833 |
ultimately show ?thesis by blast |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3834 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3835 |
from fr_eq2[OF lp, of vs bs x] show ?thesis |
55754 | 3836 |
unfolding eq0 eq1 eq2 by blast |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3837 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3838 |
|
55754 | 3839 |
definition |
55768 | 3840 |
"ferrack2 p \<equiv> |
3841 |
let |
|
3842 |
q = simpfm p; |
|
3843 |
mp = minusinf q; |
|
3844 |
pp = plusinf q |
|
3845 |
in |
|
3846 |
if (mp = T \<or> pp = T) then T |
|
3847 |
else |
|
3848 |
(let U = remdups (uset q) |
|
3849 |
in |
|
3850 |
decr0 |
|
3851 |
(list_disj |
|
3852 |
[mp, |
|
3853 |
pp, |
|
3854 |
simpfm (subst0 (CP 0\<^sub>p) q), |
|
3855 |
evaldjf (\<lambda>(c, t). msubst2 q (c *\<^sub>p C (-2, 1)) t) U, |
|
3856 |
evaldjf (\<lambda>((b, a),(d, c)). |
|
3857 |
msubst2 q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs U)]))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3858 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3859 |
definition "frpar2 p = simpfm (qelim (prep p) ferrack2)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3860 |
|
55768 | 3861 |
lemma ferrack2: |
3862 |
assumes qf: "qfree p" |
|
3863 |
shows "qfree (ferrack2 p) \<and> Ifm vs bs (ferrack2 p) = Ifm vs bs (E p)" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3864 |
(is "_ \<and> (?rhs = ?lhs)") |
55768 | 3865 |
proof - |
3866 |
let ?J = "\<lambda>x p. Ifm vs (x#bs) p" |
|
3867 |
let ?N = "\<lambda>t. Ipoly vs t" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3868 |
let ?Nt = "\<lambda>x t. Itm vs (x#bs) t" |
55754 | 3869 |
let ?q = "simpfm p" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3870 |
let ?qz = "subst0 (CP 0\<^sub>p) ?q" |
41823 | 3871 |
let ?U = "remdups(uset ?q)" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3872 |
let ?Up = "alluopairs ?U" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3873 |
let ?mp = "minusinf ?q" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3874 |
let ?pp = "plusinf ?q" |
55768 | 3875 |
fix x |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3876 |
let ?I = "\<lambda>p. Ifm vs (x#bs) p" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3877 |
from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" . |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3878 |
from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" . |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3879 |
from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" . |
55768 | 3880 |
from uset_l[OF lq] |
3881 |
have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3882 |
by simp |
55754 | 3883 |
have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)" |
55768 | 3884 |
proof - |
3885 |
{ |
|
3886 |
fix c t |
|
3887 |
assume ct: "(c, t) \<in> set ?U" |
|
3888 |
then have tnb: "tmbound0 t" |
|
3889 |
using U_l by blast |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3890 |
from msubst2_nb[OF lq tnb] |
55768 | 3891 |
have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp |
3892 |
} |
|
3893 |
then show ?thesis by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3894 |
qed |
55768 | 3895 |
have bnd1: "\<forall>x \<in> set ?Up. bound0 ((\<lambda>((b, a), (d, c)). |
3896 |
msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) x)" |
|
3897 |
proof - |
|
3898 |
{ |
|
3899 |
fix b a d c |
|
3900 |
assume badc: "((b,a),(d,c)) \<in> set ?Up" |
|
55754 | 3901 |
from badc U_l alluopairs_set1[of ?U] |
55768 | 3902 |
have nb: "tmbound0 (Add (Mul d a) (Mul b c))" |
3903 |
by auto |
|
3904 |
from msubst2_nb[OF lq nb] |
|
3905 |
have "bound0 ((\<lambda>((b, a),(d, c)). |
|
3906 |
msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) ((b,a),(d,c)))" |
|
3907 |
by simp |
|
3908 |
} |
|
3909 |
then show ?thesis by auto |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3910 |
qed |
55768 | 3911 |
have stupid: "bound0 F" |
3912 |
by simp |
|
3913 |
let ?R = |
|
3914 |
"list_disj |
|
3915 |
[?mp, |
|
3916 |
?pp, |
|
3917 |
simpfm (subst0 (CP 0\<^sub>p) ?q), |
|
3918 |
evaldjf (\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) ?U, |
|
3919 |
evaldjf (\<lambda>((b,a),(d,c)). |
|
3920 |
msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs ?U)]" |
|
3921 |
from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf |
|
3922 |
evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid |
|
3923 |
have nb: "bound0 ?R" |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
3924 |
by (simp add: list_disj_def simpfm_bound0) |
55768 | 3925 |
let ?s = "\<lambda>((b, a),(d, c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))" |
3926 |
||
3927 |
{ |
|
3928 |
fix b a d c |
|
3929 |
assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U" |
|
55754 | 3930 |
from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3931 |
by auto (simp add: isnpoly_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3932 |
have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3933 |
using norm by (simp_all add: polymul_norm) |
55768 | 3934 |
have stupid: |
3935 |
"allpolys isnpoly (CP (C (-2, 1) *\<^sub>p b *\<^sub>p d))" |
|
3936 |
"allpolys isnpoly (CP (C (-2, 1) *\<^sub>p d *\<^sub>p b))" |
|
3937 |
"allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p b *\<^sub>p d)))" |
|
3938 |
"allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p d*\<^sub>p b)))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3939 |
by (simp_all add: polyneg_norm norm2) |
55768 | 3940 |
have "?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) = |
3941 |
?I (msubst2 ?q (C (-2, 1) *\<^sub>p d*\<^sub>p b) (Add (Mul b c) (Mul d a)))" |
|
3942 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3943 |
proof |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3944 |
assume H: ?lhs |
55768 | 3945 |
then have z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
3946 |
by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] |
|
3947 |
mult_less_0_iff zero_less_mult_iff) |
|
3948 |
from msubst2[OF lq norm2(1) z(1), of x bs] msubst2[OF lq norm2(2) z(2), of x bs] H |
|
3949 |
show ?rhs |
|
3950 |
by (simp add: field_simps) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3951 |
next |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3952 |
assume H: ?rhs |
55768 | 3953 |
then have z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" |
3954 |
by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] |
|
3955 |
mult_less_0_iff zero_less_mult_iff) |
|
3956 |
from msubst2[OF lq norm2(1) z(1), of x bs] msubst2[OF lq norm2(2) z(2), of x bs] H |
|
3957 |
show ?lhs |
|
3958 |
by (simp add: field_simps) |
|
3959 |
qed |
|
3960 |
} |
|
3961 |
then have th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))" |
|
3962 |
by auto |
|
3963 |
||
3964 |
have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" |
|
3965 |
by simp |
|
3966 |
also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> |
|
3967 |
(\<exists>(n, t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> |
|
3968 |
(\<exists>(b, a) \<in> set ?U. \<exists>(d, c) \<in> set ?U. |
|
3969 |
?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3970 |
using fr_eq_msubst2[OF lq, of vs bs x] by simp |
55768 | 3971 |
also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> |
3972 |
(\<exists>(n, t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> |
|
3973 |
(\<exists>x \<in> set ?U. \<exists>y \<in>set ?U. ?I (?s (x, y)))" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3974 |
by (simp add: split_def) |
55768 | 3975 |
also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> |
3976 |
(\<exists>(n, t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> |
|
3977 |
(\<exists>(x, y) \<in> set ?Up. ?I (?s (x, y)))" |
|
55754 | 3978 |
using alluopairs_bex[OF th0] by simp |
3979 |
also have "\<dots> \<longleftrightarrow> ?I ?R" |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3980 |
by (simp add: list_disj_def evaldjf_ex split_def) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3981 |
also have "\<dots> \<longleftrightarrow> ?rhs" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3982 |
unfolding ferrack2_def |
55754 | 3983 |
apply (cases "?mp = T") |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3984 |
apply (simp add: list_disj_def) |
55754 | 3985 |
apply (cases "?pp = T") |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3986 |
apply (simp add: list_disj_def) |
55768 | 3987 |
apply (simp_all add: Let_def decr0[OF nb]) |
3988 |
done |
|
55754 | 3989 |
finally show ?thesis using decr0_qf[OF nb] |
55768 | 3990 |
by (simp add: ferrack2_def Let_def) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3991 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3992 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
3993 |
lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)" |
55754 | 3994 |
proof - |
3995 |
from ferrack2 |
|
3996 |
have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack2 p) \<and> Ifm vs bs (ferrack2 p) = Ifm vs bs (E p)" |
|
3997 |
by blast |
|
3998 |
from qelim[OF th, of "prep p" bs] |
|
55768 | 3999 |
show ?thesis |
4000 |
unfolding frpar2_def by (auto simp add: prep) |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4001 |
qed |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4002 |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4003 |
oracle frpar_oracle = {* |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4004 |
let |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4005 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4006 |
fun binopT T = T --> T --> T; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4007 |
fun relT T = T --> T --> @{typ bool}; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4008 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4009 |
val mk_C = @{code C} o pairself @{code int_of_integer}; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4010 |
val mk_poly_Bound = @{code poly.Bound} o @{code nat_of_integer}; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4011 |
val mk_Bound = @{code Bound} o @{code nat_of_integer}; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4012 |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4013 |
val dest_num = snd o HOLogic.dest_number; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4014 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4015 |
fun try_dest_num t = SOME ((snd o HOLogic.dest_number) t) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4016 |
handle TERM _ => NONE; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4017 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4018 |
fun dest_nat (t as Const (@{const_name Suc}, _)) = HOLogic.dest_nat t |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4019 |
| dest_nat t = dest_num t; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4020 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4021 |
fun the_index ts t = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4022 |
let |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4023 |
val k = find_index (fn t' => t aconv t') ts; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4024 |
in if k < 0 then raise General.Subscript else k end; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4025 |
|
55768 | 4026 |
fun num_of_term ps (Const (@{const_name Groups.uminus}, _) $ t) = |
4027 |
@{code poly.Neg} (num_of_term ps t) |
|
4028 |
| num_of_term ps (Const (@{const_name Groups.plus}, _) $ a $ b) = |
|
4029 |
@{code poly.Add} (num_of_term ps a, num_of_term ps b) |
|
4030 |
| num_of_term ps (Const (@{const_name Groups.minus}, _) $ a $ b) = |
|
4031 |
@{code poly.Sub} (num_of_term ps a, num_of_term ps b) |
|
4032 |
| num_of_term ps (Const (@{const_name Groups.times}, _) $ a $ b) = |
|
4033 |
@{code poly.Mul} (num_of_term ps a, num_of_term ps b) |
|
4034 |
| num_of_term ps (Const (@{const_name Power.power}, _) $ a $ n) = |
|
4035 |
@{code poly.Pw} (num_of_term ps a, @{code nat_of_integer} (dest_nat n)) |
|
4036 |
| num_of_term ps (Const (@{const_name Fields.divide}, _) $ a $ b) = |
|
4037 |
mk_C (dest_num a, dest_num b) |
|
4038 |
| num_of_term ps t = |
|
4039 |
(case try_dest_num t of |
|
4040 |
SOME k => mk_C (k, 1) |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4041 |
| NONE => mk_poly_Bound (the_index ps t)); |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4042 |
|
55768 | 4043 |
fun tm_of_term fs ps (Const(@{const_name Groups.uminus}, _) $ t) = |
4044 |
@{code Neg} (tm_of_term fs ps t) |
|
4045 |
| tm_of_term fs ps (Const(@{const_name Groups.plus}, _) $ a $ b) = |
|
4046 |
@{code Add} (tm_of_term fs ps a, tm_of_term fs ps b) |
|
4047 |
| tm_of_term fs ps (Const(@{const_name Groups.minus}, _) $ a $ b) = |
|
4048 |
@{code Sub} (tm_of_term fs ps a, tm_of_term fs ps b) |
|
4049 |
| tm_of_term fs ps (Const(@{const_name Groups.times}, _) $ a $ b) = |
|
4050 |
@{code Mul} (num_of_term ps a, tm_of_term fs ps b) |
|
55754 | 4051 |
| tm_of_term fs ps t = (@{code CP} (num_of_term ps t) |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4052 |
handle TERM _ => mk_Bound (the_index fs t) |
55768 | 4053 |
| General.Subscript => mk_Bound (the_index fs t)); |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4054 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4055 |
fun fm_of_term fs ps @{term True} = @{code T} |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4056 |
| fm_of_term fs ps @{term False} = @{code F} |
55768 | 4057 |
| fm_of_term fs ps (Const (@{const_name Not}, _) $ p) = |
4058 |
@{code NOT} (fm_of_term fs ps p) |
|
4059 |
| fm_of_term fs ps (Const (@{const_name HOL.conj}, _) $ p $ q) = |
|
4060 |
@{code And} (fm_of_term fs ps p, fm_of_term fs ps q) |
|
4061 |
| fm_of_term fs ps (Const (@{const_name HOL.disj}, _) $ p $ q) = |
|
4062 |
@{code Or} (fm_of_term fs ps p, fm_of_term fs ps q) |
|
4063 |
| fm_of_term fs ps (Const (@{const_name HOL.implies}, _) $ p $ q) = |
|
4064 |
@{code Imp} (fm_of_term fs ps p, fm_of_term fs ps q) |
|
4065 |
| fm_of_term fs ps (@{term HOL.iff} $ p $ q) = |
|
4066 |
@{code Iff} (fm_of_term fs ps p, fm_of_term fs ps q) |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4067 |
| fm_of_term fs ps (Const (@{const_name HOL.eq}, T) $ p $ q) = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4068 |
@{code Eq} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q)) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4069 |
| fm_of_term fs ps (Const (@{const_name Orderings.less}, _) $ p $ q) = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4070 |
@{code Lt} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q)) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4071 |
| fm_of_term fs ps (Const (@{const_name Orderings.less_eq}, _) $ p $ q) = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4072 |
@{code Le} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q)) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4073 |
| fm_of_term fs ps (Const (@{const_name Ex}, _) $ Abs (abs as (_, xT, _))) = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4074 |
let |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4075 |
val (xn', p') = Syntax_Trans.variant_abs abs; (* FIXME !? *) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4076 |
in @{code E} (fm_of_term (Free (xn', xT) :: fs) ps p') end |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4077 |
| fm_of_term fs ps (Const (@{const_name All},_) $ Abs (abs as (_, xT, _))) = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4078 |
let |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4079 |
val (xn', p') = Syntax_Trans.variant_abs abs; (* FIXME !? *) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4080 |
in @{code A} (fm_of_term (Free (xn', xT) :: fs) ps p') end |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4081 |
| fm_of_term fs ps _ = error "fm_of_term"; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4082 |
|
55754 | 4083 |
fun term_of_num T ps (@{code poly.C} (a, b)) = |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4084 |
let |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4085 |
val (c, d) = pairself (@{code integer_of_int}) (a, b) |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4086 |
in |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4087 |
(if d = 1 then HOLogic.mk_number T c |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4088 |
else if d = 0 then Const (@{const_name Groups.zero}, T) |
55768 | 4089 |
else |
4090 |
Const (@{const_name Fields.divide}, binopT T) $ |
|
4091 |
HOLogic.mk_number T c $ HOLogic.mk_number T d) |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4092 |
end |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4093 |
| term_of_num T ps (@{code poly.Bound} i) = nth ps (@{code integer_of_nat} i) |
55768 | 4094 |
| term_of_num T ps (@{code poly.Add} (a, b)) = |
4095 |
Const (@{const_name Groups.plus}, binopT T) $ term_of_num T ps a $ term_of_num T ps b |
|
4096 |
| term_of_num T ps (@{code poly.Mul} (a, b)) = |
|
4097 |
Const (@{const_name Groups.times}, binopT T) $ term_of_num T ps a $ term_of_num T ps b |
|
4098 |
| term_of_num T ps (@{code poly.Sub} (a, b)) = |
|
4099 |
Const (@{const_name Groups.minus}, binopT T) $ term_of_num T ps a $ term_of_num T ps b |
|
4100 |
| term_of_num T ps (@{code poly.Neg} a) = |
|
4101 |
Const (@{const_name Groups.uminus}, T --> T) $ term_of_num T ps a |
|
4102 |
| term_of_num T ps (@{code poly.Pw} (a, n)) = |
|
4103 |
Const (@{const_name Power.power}, T --> @{typ nat} --> T) $ |
|
4104 |
term_of_num T ps a $ HOLogic.mk_number HOLogic.natT (@{code integer_of_nat} n) |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4105 |
| term_of_num T ps (@{code poly.CN} (c, n, p)) = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4106 |
term_of_num T ps (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p))); |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4107 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4108 |
fun term_of_tm T fs ps (@{code CP} p) = term_of_num T ps p |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50282
diff
changeset
|
4109 |
| term_of_tm T fs ps (@{code Bound} i) = nth fs (@{code integer_of_nat} i) |
55768 | 4110 |
| term_of_tm T fs ps (@{code Add} (a, b)) = |
4111 |
Const (@{const_name Groups.plus}, binopT T) $ term_of_tm T fs ps a $ term_of_tm T fs ps b |
|
4112 |
| term_of_tm T fs ps (@{code Mul} (a, b)) = |
|
4113 |
Const (@{const_name Groups.times}, binopT T) $ term_of_num T ps a $ term_of_tm T fs ps b |
|
4114 |
| term_of_tm T fs ps (@{code Sub} (a, b)) = |
|
4115 |
Const (@{const_name Groups.minus}, binopT T) $ term_of_tm T fs ps a $ term_of_tm T fs ps b |
|
4116 |
| term_of_tm T fs ps (@{code Neg} a) = |
|
4117 |
Const (@{const_name Groups.uminus}, T --> T) $ term_of_tm T fs ps a |
|
4118 |
| term_of_tm T fs ps (@{code CNP} (n, c, p)) = |
|
4119 |
term_of_tm T fs ps (@{code Add} (@{code Mul} (c, @{code Bound} n), p)); |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4120 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4121 |
fun term_of_fm T fs ps @{code T} = @{term True} |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4122 |
| term_of_fm T fs ps @{code F} = @{term False} |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4123 |
| term_of_fm T fs ps (@{code NOT} p) = @{term Not} $ term_of_fm T fs ps p |
55768 | 4124 |
| term_of_fm T fs ps (@{code And} (p, q)) = |
4125 |
@{term HOL.conj} $ term_of_fm T fs ps p $ term_of_fm T fs ps q |
|
4126 |
| term_of_fm T fs ps (@{code Or} (p, q)) = |
|
4127 |
@{term HOL.disj} $ term_of_fm T fs ps p $ term_of_fm T fs ps q |
|
4128 |
| term_of_fm T fs ps (@{code Imp} (p, q)) = |
|
4129 |
@{term HOL.implies} $ term_of_fm T fs ps p $ term_of_fm T fs ps q |
|
4130 |
| term_of_fm T fs ps (@{code Iff} (p, q)) = |
|
4131 |
@{term HOL.iff} $ term_of_fm T fs ps p $ term_of_fm T fs ps q |
|
4132 |
| term_of_fm T fs ps (@{code Lt} p) = |
|
4133 |
Const (@{const_name Orderings.less}, relT T) $ |
|
4134 |
term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T) |
|
4135 |
| term_of_fm T fs ps (@{code Le} p) = |
|
4136 |
Const (@{const_name Orderings.less_eq}, relT T) $ |
|
4137 |
term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T) |
|
4138 |
| term_of_fm T fs ps (@{code Eq} p) = |
|
4139 |
Const (@{const_name HOL.eq}, relT T) $ |
|
4140 |
term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T) |
|
4141 |
| term_of_fm T fs ps (@{code NEq} p) = |
|
4142 |
@{term Not} $ |
|
4143 |
(Const (@{const_name HOL.eq}, relT T) $ |
|
4144 |
term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)) |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4145 |
| term_of_fm T fs ps _ = error "term_of_fm: quantifiers"; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4146 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4147 |
fun frpar_procedure alternative T ps fm = |
55754 | 4148 |
let |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4149 |
val frpar = if alternative then @{code frpar2} else @{code frpar}; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4150 |
val fs = subtract (op aconv) (map Free (Term.add_frees fm [])) ps; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4151 |
val eval = term_of_fm T fs ps o frpar o fm_of_term fs ps; |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4152 |
val t = HOLogic.dest_Trueprop fm; |
55768 | 4153 |
in HOLogic.mk_Trueprop (HOLogic.mk_eq (t, eval t)) end; |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4154 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4155 |
in |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4156 |
|
55754 | 4157 |
fn (ctxt, alternative, ty, ps, ct) => |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4158 |
cterm_of (Proof_Context.theory_of ctxt) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4159 |
(frpar_procedure alternative ty ps (term_of ct)) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4160 |
|
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4161 |
end |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4162 |
*} |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4163 |
|
55754 | 4164 |
ML {* |
4165 |
structure Parametric_Ferrante_Rackoff = |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4166 |
struct |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4167 |
|
55754 | 4168 |
fun tactic ctxt alternative T ps = |
54742
7a86358a3c0b
proper context for basic Simplifier operations: rewrite_rule, rewrite_goals_rule, rewrite_goals_tac etc.;
wenzelm
parents:
54230
diff
changeset
|
4169 |
Object_Logic.full_atomize_tac ctxt |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4170 |
THEN' CSUBGOAL (fn (g, i) => |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4171 |
let |
55768 | 4172 |
val th = frpar_oracle (ctxt, alternative, T, ps, g); |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4173 |
in rtac (th RS iffD2) i end); |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4174 |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4175 |
fun method alternative = |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4176 |
let |
55768 | 4177 |
fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K (); |
4178 |
val parsN = "pars"; |
|
4179 |
val typN = "type"; |
|
4180 |
val any_keyword = keyword parsN || keyword typN; |
|
4181 |
val terms = Scan.repeat (Scan.unless any_keyword Args.term); |
|
4182 |
val typ = Scan.unless any_keyword Args.typ; |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4183 |
in |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4184 |
(keyword typN |-- typ) -- (keyword parsN |-- terms) >> |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4185 |
(fn (T, ps) => fn ctxt => SIMPLE_METHOD' (tactic ctxt alternative T ps)) |
55768 | 4186 |
end; |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4187 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4188 |
end; |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4189 |
*} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4190 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4191 |
method_setup frpar = {* |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4192 |
Parametric_Ferrante_Rackoff.method false |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4193 |
*} "parametric QE for linear Arithmetic over fields" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4194 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4195 |
method_setup frpar2 = {* |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4196 |
Parametric_Ferrante_Rackoff.method true |
42814 | 4197 |
*} "parametric QE for linear Arithmetic over fields, Version 2" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4198 |
|
55768 | 4199 |
lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1) * x < 0" |
4200 |
apply (frpar type: 'a pars: y) |
|
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
4201 |
apply (simp add: field_simps) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4202 |
apply (rule spec[where x=y]) |
55768 | 4203 |
apply (frpar type: 'a pars: "z::'a") |
55754 | 4204 |
apply simp |
4205 |
done |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4206 |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4207 |
lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0" |
55768 | 4208 |
apply (frpar2 type: 'a pars: y) |
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4209 |
apply (simp add: field_simps) |
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4210 |
apply (rule spec[where x=y]) |
55768 | 4211 |
apply (frpar2 type: 'a pars: "z::'a") |
55754 | 4212 |
apply simp |
4213 |
done |
|
50045
2214bc566f88
modernized, simplified and compacted oracle and proof method glue code;
haftmann
parents:
49962
diff
changeset
|
4214 |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4215 |
text{* Collins/Jones Problem *} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4216 |
(* |
36409 | 4217 |
lemma "\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4218 |
proof- |
36409 | 4219 |
have "(\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs") |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
4220 |
by (simp add: field_simps) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4221 |
have "?rhs" |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4222 |
|
36409 | 4223 |
apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "a::'a::{linordered_field_inverse_zero, number_ring}" "b::'a::{linordered_field_inverse_zero, number_ring}") |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
4224 |
apply (simp add: field_simps) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4225 |
oops |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4226 |
*) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4227 |
(* |
36409 | 4228 |
lemma "ALL (x::'a::{linordered_field_inverse_zero, number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y" |
4229 |
apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}") |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4230 |
oops |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4231 |
*) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4232 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4233 |
text{* Collins/Jones Problem *} |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4234 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4235 |
(* |
36409 | 4236 |
lemma "\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0" |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4237 |
proof- |
36409 | 4238 |
have "(\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs") |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
35625
diff
changeset
|
4239 |
by (simp add: field_simps) |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4240 |
have "?rhs" |
36409 | 4241 |
apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "a::'a::{linordered_field_inverse_zero, number_ring}" "b::'a::{linordered_field_inverse_zero, number_ring}") |
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4242 |
apply simp |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4243 |
oops |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4244 |
*) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4245 |
|
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4246 |
(* |
36409 | 4247 |
lemma "ALL (x::'a::{linordered_field_inverse_zero, number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y" |
4248 |
apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}") |
|
33152
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4249 |
apply (simp add: field_simps linorder_neq_iff[symmetric]) |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4250 |
apply ferrack |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4251 |
oops |
241cfaed158f
Add a quantifier elimination for parametric linear arithmetic over ordered fields (parameters are multivariate polynomials)
chaieb
parents:
diff
changeset
|
4252 |
*) |
45499
849d697adf1e
Parametric_Ferrante_Rackoff.thy: restrict to class number_ring, replace '1+1' with '2' everywhere
huffman
parents:
44064
diff
changeset
|
4253 |
end |